Gelfand Semirings, m-semirings and the Zariski Topology
|
|
- Jayson Webster
- 5 years ago
- Views:
Transcription
1 International Journal of Algebra, Vol. 3, 2009, no. 20, Gelfand Semirings, m-semirings and the Zariski Topology L. M. Ruza Departamento de Matemáticas, Facultad de Ciencias Universidad de los Andes, Merida, Venezuela J. Vielma Departamento de Matemáticas, Facultad de Ciencias Universidad de los Andes, Merida, Venezuela Abstract In this work by a semiring R we understand a commutative semiring with identity and we consider in its prime spectrum the Zariski topology t z.we denote by t z the smallest Alexandroff topology containing t z. Also t z denotes its corresponding cotopology. We say that R is a Gelfand semiring if every prime ideal is contained in a unique maximal ideal. We say that R is an m-semiring if each prime ideal contains only one minimal prime ideal.we give a characterization of such semirings in terms of the clopen subsets of the smallest Alexandroff topology t z in Spec(R) containing the Zariski topology t z. Also, characterizations of the compactness and connectedness of the spectrum of such semirings, in terms of the topologies t z and t z, are given. Mathematics Subject Classification: 54E18, 54F65, 13C05 Keywords: Alexandroff topology, Zariski topology, Gelfand semirings, m- semirings, prime spectrum 1 Introduction Let R be a commutative semiring with non-zero identity. Spec(R) denotes the set of all prime ideals of R, equipped with the Zariski topology t z ([7]). For every proper set I of R, we denote by (I) 0 the set of all prime ideals of R containing I, and D 0 (I) =Spec(R) (I) 0. Max(R) and Min(R) denote
2 982 L. M. Ruza and J. Vielma the set of all maximal and minimal prime ideals of R, respectively. If we view t z as a subset of 2 Spec(R) with the product topology, then its closure t z is also a topology and it is the smallest Alexandroff topology containing t z ([8]). A topology is said to be Alexandroff if it is closed under arbitrary intersections. For any point x in a topological space (X, τ), Ker(x) will denote the intersection of all τ-open sets containing x. Also A τ is a τ-closed set and is just the union of the τ-closure of each of its points. We say that R is a Gelfand semiring if every prime ideal is contained in a unique maximal ideal. Following Avila in [3], we say that R is an m-semiring if each prime ideal contains only one minimal prime ideal. The Gelfand rings were characterized in [6], as those rings in which the maximal prime spectrum is a retract of (Spec(R),t z ). We extend this result to the more general case when R is a Gelfand semiring. In addition, we show that R is a Gelfand semiring if and only if for every M in Max(R), Ker(M) is clopen in t z. We show that R is an m-semiring if and only if for every prime ideal M Min(R), (M) 0 is clopen in t z if and only if Min(R) is a retract of (Spec(R), t z ). We give a characterization of the compactness and connectedness of the spectrun of such semirings in terms of the topologies t z and t z. A semiring R is said to be semilocal if it has a finite number of maximal ideals. R is called local if it has only one maximal ideal. A topological space (X, τ) is said to be nearly compact (almost compact) if every τ-open cover Θ of X contains a finite subfamily {U i : i = 1...n} such that X = n i=1 intu i (X = n i=1 U i). In fact, we prove that for Gelfand semirings, t z -compactness, t z -nearly compactness and t z -almost compactness are equivalent to the fact that R is a semilocal semiring. Also for m-semirings, t z -compactness, t z -nearly compactness and t z -almost compactness are also equivalent to the condition that R has a finite number of minimal prime ideals. 2 Terminology On the following, N := {0, 1, 2,...} denotes the set of natural numbers. Also, by a space we understand a topological space, (X, τ) always denotes a space and τ the family of τ-closed subsets of X ([8]). If A 2 X, we denote by A the intersection of all the elements of A. Recall that a semiring (commutative with non-zero identity) is an algebra (R, +,, 0, 1), where R is a set with 0, 1 R, and + and are binary operations on R called sum and multiplication, respectively, which satisfy the following: (S 1 )(R, +, 0) and (R,, 1) are commutative monoids with 1 0. (S 2 ) a (b + c) =a b + a c for every a, b, c R.
3 Gelfand semirings, m-semirings and the Zariski topology 983 (S 3 ) a 0 = 0 for every a R. As is usual, we denote a semiring (R, +,, 0, 1) by R. The notions of (proper) ideal, prime ideal and maximal ideal of a semiring R are defined as in commutative rings ([7]). Example 2.1 The set N with the usual addition and multiplication is a semiring, which it has an unique maximal ideal M = N {1} ([7]). Example 2.2 The set Q + of all nonnegative rational numbers, with the usual addition and multiplication is a semiring; the same is true for the set R + of all nonnegative real numbers ([7]). Example 2.3 If (X, τ) is a topological space, then τ is a semiring with operations of addition and multiplication given by A + B = A B and AB = A B. The additive identity is and multiplicative identity X. (Easy verification). Example 2.4 Let R = R { }. Then (R, min, +) is a commutative semiring in which addition is the operation of taking minimum and multiplication is ordinary addition. This semiring is important in solving the shortest-path problem in optimization (example 1.22 in [7]). Example 2.5 If X is a Hausdorff topological space then the set R of all continuous bounded functions from X to R + is a commutative semiring ([5]). The following examples are discussed with great detail in [1]. 3 The Semirings B(n, i) Let R be a semiring with multiplicative identity 1 R and additive identity 0 R. The set N1 ={n1 R : n N} is a commutative subsemiring of R, where 0 R = 0.1 R N1 1 R = 1.1 R N1 n1 R + m1 R = (n + m)1 R N1. n1 R.m1 R = (nm)1 R N1. Let us consider the case when k1 R 0 R for k>1 and a1 R = b1 R for some a b. Let n be the least positive integer with n1 R = i1 R where 1 i n 1. Here we write j for j1 R so N1 ={0, 1, 2,..., (n 1)}. It such case, if m = n i the followings holds:
4 984 L. M. Ruza and J. Vielma a + b = { a + b, if 0 a + b n 1 l, with l a + b mod m, if a + b n. where, l is the unique number such that l a + b modm with i l n 1. And the multiplication { ab, if 0 ab n 1 ab = l, with l ab mod m, if ab n. (l is the unique number such that l ab modm with i l n 1). Conversely, given n 2 and 0 i n 1 there is a unique (up to isomorphism) semiring of the kind described above. Let B(n, i) ={0, 1,.., n 1} and put m = n i. Make B(n, i) into a semiring by defining the operation of addition as following a + b = { a + b, si 0 a + b n 1 l, con l a + b mod m, si a + b n. then l is the unique number such that l a + b modm with i l n 1. The product is defined similarly. The following theorem, which appears in [1] as Theorem 24, is included so that the reader can appreciate some of the properties of such semirings. Theorem 3.1 (1) dimb(n, i) =0if i =0or i =1and n =2. (2) dimb(n, i) =1if i =1and n>2. In this case the prime ideals of B(n, i) are 0 and pb(n, i) where p is a prime with p n 1. (3) dimb(n, i) =1if n>2 and n = i +1. In this case the prime ideals of B(n, i) are 0 and {0, 2, 3,..., n 1}. (4) dimb(n, i) =2si n 1 >i 2. In this case the prime ideals are 0, M = {0, 2, 3,..., n 1} and pb(n, i) where p is prime and p n i. 4 Gelfand Semirings Remember that a semiring R is a Gelfand semiring if each prime ideal is contained in a unique maximal prime ideal. Example 4.1 The nonnegative integers N, with the usual addition and multiplication, is an example of a Gelfand semiring. More than that, N is a local semiring whose only maximal ideal is M = N {1}. For more details see ([1]). A good list of Gelfand semirings is the one that comes out of (4) in Theorem 3.1.
5 Gelfand semirings, m-semirings and the Zariski topology 985 Lemma 4.1 If A Spec(R), then A = {P Spec(R) : A P } Proof. First observe that B = {P Spec(R) : A P } is a closed subset of Spec(R), since B =( A) 0. Also, A B. Then, A B. Now, since A is closed in Spec(R) we have that A =(I) 0 for some ideal I. Take P Spec(R) such that A P. Then since A A =(I) 0, it follows that I A P. So, P (I) 0. Therefore, P A. Lemma 4.2 Let μ : Spec(R) Max(R), be the function defined by μ(p )= M P, where M P is the unique maximal ideal that contains P, when R be Gelfand semiring. If D be a closed subset of Max(R), F = D, Q Spec(R) and Q B = {M : M D}, then μ(q) D. Proof. Since D is closed in Max(R), there exist a closed set A in Spec(R) such that D = A Max(R). Also, A D. On the other hand, Q + F B, then there exist a maximal ideal M such that Q + F M. Also, since F M, then A F,soM A Max(R) which implies that M D. Then μ(q) D.. What follows, is the version for semirings of the well known Krull s Lemma [2], which will be stated without proof. Lemma 4.3 (Krull) Let S be a multiplicative closed subset of a semiring R and I an ideal of R such that I S = φ. Then, there exists an ideal P of R maximal with respect to the property P S = φ and I P. Further, every such an ideal is prime. The following theorem is well known for commutative rings [6]. Theorem 4.1 A semiring R is Gelfand if and only if Max(R) is a retract of (Spec(R),t z ). Proof. Let μ : Spec(R) Max(R), be the function defined by μ(p )=M P, where M P is the unique maximal ideal that contains P. We show that μ is continuous. Let D be a closed subset of Max(R) and consider F = D and I = {P Spec(R) :μ(p ) D}. Let P Spec(R) and I P. We show that P contains a prime ideal Q B = {M : M D}, which implies that μ(p )=μ(q) D by Lemma 4.2. Let S = R B and T = R P, choose s S, t T. Since I P, there exist P μ 1 (D) such that t/ P. Also, since s/ P, it follows that st / P. Therefore, st / I. That means that the closed multiplicative system ST does not intersect I. By Krull s Lemma, there exists a prime ideal Q containing I, and disjoint from ST, Since Q B and Q P, it follows that μ(p )=μ(q) D. So, μ is continuous.
6 986 L. M. Ruza and J. Vielma Conversely, if φ is a retract from Spec(R)ontoMax(R). Take P Spec(R) with φ(p )=M. Then, P φ 1 ({M}). Since every maximal prime ideal is a closed point, we have that φ 1 ({M}) is closed in Spec(R). Then, {P } φ 1 ({M}). Therefore, if M 1 Max(R) and P M 1 we have that M 1 {P } =(P ) 0. So, M 1 = φ(m 1 )=M, R is Gelfand. Lemma 4.4 Let R be a semiring. If M is a prime ideal of R and Ker(M) is t z -clopen, then M Max(R). Proof. Suppose P is a prime ideal of R with Ker(P ) t z -clopen. Then (P ) 0 Ker(P ). Since t z is a T 0 topology, we have that {P } =(P ) 0 Ker(P ), and then {P } =(P ) 0 which implies that P is maximal. Theorem 4.2 A semiring R es Gelfand if and only if for every M in Max(R), Ker(M) is clopen in t z. Proof. Let M be a maximal prime. Since Ker(M) ist z -open, it remains only to prove that Ker(M) ist z -closed. Let P Ker(M) and we see that (P ) 0 Ker(M). Let Q (P ) 0, then P Q. If M Q is the only maximal prime containing Q, it follows that M Q = M. So, Q Ker(M). Conversely, if P is a prime ideal and M 1, M 2 are maximal ideals containing P. Then, P Ker(M 1 ) and P Ker(M 2 ). By hypothesis, we have that (P ) 0 Ker(M 1 ) and (P ) 0 Ker(M 2 ). So, M 1 Ker(M 2 ) and therefore M 1 = M 2. Then R is a Gelfand semiring. Remember that a semiring R is said to be semilocal if it has a finite number of maximal ideals. R is called local if it has only one maximal ideal. Theorem 4.3 Let R be a Gelfand semiring. The following are equivalent. (a) (Spec(R), t z ) is compact (b) (Spec(R), t z ) is nearly-compact (c) (Spec(R), t z ) is almost-compact (d) R is a semilocal semiring (e) (Spec(R), t z t z ) is compact Proof. Is obvious that (a) (b), (b) (c). By Lemma 4.1, (c) (d). Let us prove that (d) (a). Let {U α } a covering by t z -open subsets of Spec(R). By (d), there exits a finite number of maximal ideals, say M 1,.., M k. Then for each M i there exits an U αi such that M i U αi. Then, Ker(M i ) U αi and {U αi } is a finite subcover of Spec(R). Clearly (a) implies (e). Now, If {U α } is a covering by t z -open subsets of Spec(R), for each M Max(R) there exits
7 Gelfand semirings, m-semirings and the Zariski topology 987 an U αm such that M U αm. Then, Ker(M) U αm. Now, since R is Gelfand, it follows that {Ker(M),M Max(R)} is an t z t z open covering of Spec(R), so there is a finite subcovering {Ker(M i ):i =1,..., n, M i Max(R)} and then {U αi : i =1,..., n} is a finite subcover of Spec(R). Proposition 4.1 If R is a Gelfand semiring and (Spec(R), t z ) is connected, then R is a local semiring. Proof. Let M be a maximal ideal in R, then Ker(M) ist z -clopen. Since Ker(M), it follows that Spec(R) =Ker(M). Therefore, M is the unique maximal ideal of R. Lemma 4.5 If R is a local semiring, then (Spec(R), t z ) is connected. Proof. Let M be the unique maximal ideal of R. and Ω a nonempty τ z -clopen subset of Spec(R). If P U, then (P ) 0 U, and then M U y Ker(M) U. Now since Ker(M) = Spec(R), it follows that U = Spec(R). Let us remember that two topologies on a space X are said to be complementary if their supremum is the discrete topology and their intersection is the indiscrete one. Theorem 4.4 Let R be a Gelfand semiring. The following are equivalent: (a) (Spec(R), t z ) is connected. (b) R is a local semiring. (c) t z y t z are complementary topologies. Proof. Proposition 4.1 and Lemma 4.5 implies that (a) and (b) are equivalent. Clearly (c) implies (a). Now, since t 0 is a T 0 topology [9], then the supremum of t z and t z is the discrete topology. Also, since t z is connected then the intersection of t z and t z is the indiscrete topology. 5 m-semirings Remember that a semiring R is an m-semiring if each prime ideal contains only one minimal prime ideal. Example 5.1 The nonnegative integers N, with the usual addition and multiplication, is also a good example of an m-semiring which is not an m-ring. More than that, N is a local semiring with only one minimal ideal. For more details see [1].An extense list of m-semirings can be found if if we look at the examples that come out from (2) in Theorem 3.1.
8 988 L. M. Ruza and J. Vielma Remember that a topological space X is supercompact if X belongs to every open cover of X. The supercompact elements in complete lattices are introduced in [4]. Theorem 5.1 (Spec(R), t z) is supercompact if and only if R has only one minimal prime ideal. Proof. Let Π = {(M) 0 : M Min(R)}. Then Π is an t z-open covering of Spec(R). It follows that for some (M) 0 Π, Spec(R) =(M) 0. Therefore Min(R) (M) 0, which implies that M is the unique minimal prime ideal of R. Conversely, if Π is an t z-open covering of Spec(R), then the unique minimal prime ideal M of R belongs to one of the elements in Π, say A M. But A M = N A M (N) 0. That implies that M (M ) 0 for some M A M. Therefore A M = Spec(R), which proves that (Spec(R), t z ) is supercompact. Lemma 5.1 Let R be a semiring. If M is a prime ideal of R and (M) 0 is t z -clopen, then M Min(R). Proof. Let P be a prime ideal and (P ) 0 t z -clopen. Then Ker(P ) (P ) 0. Since Since t z is a T 0 topology, {P } =(P ) 0 Ker(P ), then {P } = Ker(P ) which implies that P is minimal. Theorem 5.2 A semiring R is an m-semiring if and only if for each minimal prime ideal M, (M) 0 is clopen in t z. Proof. Let M be a minimal prime. Since (M) 0 is t z -closed we only need to prove that (M) 0 es kernelled. Let P (M) 0 and take Q Ker(P ), then Q P. If M Q is the unique minimal ideal contained in Q it follows that M Q P. Therefore M Q is the unique minimal ideal contained in P. Then M Q = M and Q (M) 0. Conversely, take P prime and M 1, M 2 minimal prime ideals contained in P. Since (M 1 ) 0 and (M 2 ) 0 are clopen in t z and P (M 1 ) 0, it follows that Ker(P ) (M 1 ) 0. Also P (M 2 ) 0, therefore M 2 (M 1 ) 0. Then M 1 = M 2, which completes the proof. Lemma 5.2 Let R be a semiring. A prime ideal P of R is minimal if and only if Ker(P )={P }. Proof. It is trivial Theorem 5.3 Let R be a semiring. R is an m-semiring if and only if Min(R) is a retract of (Spec(R), t z ).
9 Gelfand semirings, m-semirings and the Zariski topology 989 Proof. Suppose φ be a retract from Spec(R) ontomin(r). Let P be a prime ideal, M 1 and M 2 minimal prime ideals contained in P. Since {M 1 } is kernelled in Spec(R), then {M 1 } is kernelled in Min(R) and therefore φ 1 ({M 1 })isa kernelled subset of Spec(R) which contains P. Then it follows that Ker(P )isa subset of φ 1 ({M 1 }) and that M 2 belongs to φ 1 ({M 1 }). Therefore M 1 = M 2 and R is an m-semiring. Conversely, If R is an m-semiring, the map φ : Spec(R) Min(R) defined by φ(p )=m P, where m P is the unique minimal prime ideal contained in P, is well defined and from Theorem 5.2 and Lemma 5.2 it is easy to show that φ is continuous. Lemma 5.3 Let R be an m-semiring. If (Spec(R), t z ) is almost compact then there exists a finite number of minimal prime ideals in the semiring. Proof. Let us consider the covering {(m) 0 : m Min(R)} by t z -open subsets of Spec(R). Then there exits a finite subcollection {(m i ) 0 : i =1,..., n} such that Spec(R) = (m i ) 0 = (m i ) 0. Therefore, the only minimal prime ideals of the semiring are {m i }. Theorem 5.4 Let R be an m-semiring. The following are equivalent. (a) (Spec(R), t z ) is compact (b) (Spec(R), t z ) is nearly-compact (c) (Spec(R), t z ) is almost-compact (d) R has a finite number of minimal prime ideals (e) (Spec(R), t z t z ) is compact Proof. The following implications are trivial: (a) (b), (b) (c), and by Lemma??, (c) (d). Let see that (d) (a). Let {U α } be an t z -open covering of Spec(R). For each minimal ideal M there exists an U αm with M U αm. Since (M) o is a subset of {U αm }, and the fact that {(M) o : M Min(R)} is an (t z t z )-open covering of Spec(R), then by (d), there exist a finite number of minimal prime ideals, say M 1,...,M k such that (M 1 ) o,...,(m k ) 0 covers Spec(R). So, {U αmi } is a finite subcovering of Spec(R). Proposition 5.1 If R is an m-semiring and (Spec(R), t z ) is connected, then there exits a unique minimal prime ideal. Proof. It follows form the fact that (m) 0 is a t z -clopen if m is a minimal prime ideal. Lemma 5.4 If R is a semiring with a unique minimal prime ideal, then (Spec(R), t z ) is connected.
10 990 L. M. Ruza and J. Vielma Proof. Let m be the unique minimal prime ideal of R and take U Spec(R) a nontrivial t z -clopen subset. If m U, then (m) 0 U and since (m) 0 = Spec(R) we get a contradiction. Similarly if m/ U. Remember that a space is irreducible if every open subset is dense. Lemma 5.5 Let R be a semiring. (Spec(R),t z ) is irreducible if and only if η(0) is a prime ideal. In such case, η(0) is the only minimal prime ideal. Proof. Let ab η(0), then ab P for every P Spec(R). Take P and Q Spec(R) such that a/ P y b/ Q. Then, P D 0 (a) yq D 0 (b), by hypothesis there exist W D 0 (a) D 0 (b). Then, a/ W y b/ W, contradicting the fact that ab η(0) W. Conversely, if η(0) is a prime ideal, let P D 0 (a) yq D 0 (b). Since, η(0) P D 0 (a) and η(0) Q D 0 (b). We get that D 0 (a) D 0 (b). The last part is evident. Theorem 5.5 Let R be a semiring. The following are equivalent: (a) (Spec(R), t z ) is connected. (b) R has a unique minimal prime ideal. (c) (Spec(R),t z ) is an irreducible space. (d) η(0) is the unique minimal prime ideal. (e) t z y t z are complementary topologies. Proof. Proposition 5.1 and lemma 5.4 implies that (a) and (b) are equivalent. Lemma 5.5 implies the equivalence of (c) and (d). Since t z is a T 0 topology and t z is connected, then (a) and (d) are equivalent. The equivalence of (b) and (d) is trivial. ACKNOWLEDGEMENTS: We are grateful to Dr Daniel Anderson for his comments and suggestions. References [1] F. Alarcón, D.D. Anderson, Commutative semirings and their lattices of ideals, Houston J. Math. 20 (4) (1994), [2] M. Atiyah, I. Macdonald, Introduction to commutative algebra, Addison- Wesley, P.C., 1969.
11 Gelfand semirings, m-semirings and the Zariski topology 991 [3] J. A, Ávila, Spec(R) y Axiomas de Separacion entre T 0 y T 1, Divulgaciones Matematicas. 13 (2)(2005), [4] B. Banaschewski, S.B. Niefield, Proyective and supercoherent frames, J. Pure Appl. Algebra 70 (1991), [5] K. Iséki, Y. Miyanaga, Notes on topological spaces IV. Function semiring on topological spaces, Proc. Japan. Acad. 32 (1956), [6] G. De Marco, A. Orsatti, Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Amer. Math. Soc. 30(3)(1971), [7] J. S. Golan, Semirings and their applications, Kluwer Academic Publishers, Dordrecht (1999). [8] C. Uzcátegui, J. Vielma, Alexandroff Topologies viewed as closed subsets of the Cantor cube, Divulg. Mat. 13 (1)(2005),45-53 [9] M.L.Colasante, C. Uzcátegui,and J. Vielma, Boolean algebras and low separation axioms,(to appear in Topology Proceedings, 2008). Received: October, 2008
P-Spaces and the Prime Spectrum of Commutative Semirings
International Mathematical Forum, 3, 2008, no. 36, 1795-1802 P-Spaces and the Prime Spectrum of Commutative Semirings A. J. Peña Departamento de Matemáticas, Facultad Experimental de Ciencias, Universidad
More informationClasses of Commutative Clean Rings
Classes of Commutative Clean Rings Wolf Iberkleid and Warren Wm. McGovern September 3, 2009 Abstract Let A be a commutative ring with identity and I an ideal of A. A is said to be I-clean if for every
More informationBoolean Algebras, Boolean Rings and Stone s Representation Theorem
Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to
More informationThe Space of Maximal Ideals in an Almost Distributive Lattice
International Mathematical Forum, Vol. 6, 2011, no. 28, 1387-1396 The Space of Maximal Ideals in an Almost Distributive Lattice Y. S. Pawar Department of Mathematics Solapur University Solapur-413255,
More informationTHE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter
THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS K. R. Goodearl and E. S. Letzter Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations,
More informationIdeals Of The Ring Of Higher Dimensional Dual Numbers
Journal of Advances in Algebra (AA). ISSN 0973-6964 Volume 9, Number 1 (2016), pp. 1 8 Research India Publications http://www.ripublication.com/aa.htm Ideals Of The Ring Of Higher Dimensional Dual Numbers
More informationIntroduction to generalized topological spaces
@ Applied General Topology c Universidad Politécnica de Valencia Volume 12, no. 1, 2011 pp. 49-66 Introduction to generalized topological spaces Irina Zvina Abstract We introduce the notion of generalized
More informationz -FILTERS AND RELATED IDEALS IN C(X) Communicated by B. Davvaz
Algebraic Structures and Their Applications Vol. 2 No. 2 ( 2015 ), pp 57-66. z -FILTERS AND RELATED IDEALS IN C(X) R. MOHAMADIAN Communicated by B. Davvaz Abstract. In this article we introduce the concept
More informationJ. Sanabria, E. Acosta, M. Salas-Brown and O. García
F A S C I C U L I M A T H E M A T I C I Nr 54 2015 DOI:10.1515/fascmath-2015-0009 J. Sanabria, E. Acosta, M. Salas-Brown and O. García CONTINUITY VIA Λ I -OPEN SETS Abstract. Noiri and Keskin [8] introduced
More informationON SOME VERY STRONG COMPACTNESS CONDITIONS
Acta Math. Hungar., 130 (1 2) (2011), 188 194 DOI: 10.1007/s10474-010-0007-9 First published online November 3, 2010 ON SOME VERY STRONG COMPACTNESS CONDITIONS M. GANSTER 1,S.JAFARI 2 and M. STEINER 1
More informationMaximilian GANSTER. appeared in: Soochow J. Math. 15 (1) (1989),
A NOTE ON STRONGLY LINDELÖF SPACES Maximilian GANSTER appeared in: Soochow J. Math. 15 (1) (1989), 99 104. Abstract Recently a new class of topological spaces, called strongly Lindelöf spaces, has been
More informationA NOTE ON Θ-CLOSED SETS AND INVERSE LIMITS
An. Şt. Univ. Ovidius Constanţa Vol. 18(2), 2010, 161 172 A NOTE ON Θ-CLOSED SETS AND INVERSE LIMITS Ivan Lončar Abstract For every Hausdorff space X the space X Θ is introduced. If X is H-closed, then
More informationRings With Topologies Induced by Spaces of Functions
Rings With Topologies Induced by Spaces of Functions Răzvan Gelca April 7, 2006 Abstract: By considering topologies on Noetherian rings that carry the properties of those induced by spaces of functions,
More informationTotally supra b continuous and slightly supra b continuous functions
Stud. Univ. Babeş-Bolyai Math. 57(2012), No. 1, 135 144 Totally supra b continuous and slightly supra b continuous functions Jamal M. Mustafa Abstract. In this paper, totally supra b-continuity and slightly
More informationTROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),
More informationCongruence Boolean Lifting Property
Congruence Boolean Lifting Property George GEORGESCU and Claudia MUREŞAN University of Bucharest Faculty of Mathematics and Computer Science Academiei 14, RO 010014, Bucharest, Romania Emails: georgescu.capreni@yahoo.com;
More informationNotes about Filters. Samuel Mimram. December 6, 2012
Notes about Filters Samuel Mimram December 6, 2012 1 Filters and ultrafilters Definition 1. A filter F on a poset (L, ) is a subset of L which is upwardclosed and downward-directed (= is a filter-base):
More informationON µ-compact SETS IN µ-spaces
Questions and Answers in General Topology 31 (2013), pp. 49 57 ON µ-compact SETS IN µ-spaces MOHAMMAD S. SARSAK (Communicated by Yasunao Hattori) Abstract. The primary purpose of this paper is to introduce
More informationExploring the Exotic Setting for Algebraic Geometry
Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 6-10, 2010 1 Introduction In this project, we will describe the basic topology
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationTHE NON-URYSOHN NUMBER OF A TOPOLOGICAL SPACE
THE NON-URYSOHN NUMBER OF A TOPOLOGICAL SPACE IVAN S. GOTCHEV Abstract. We call a nonempty subset A of a topological space X finitely non-urysohn if for every nonempty finite subset F of A and every family
More informationNAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key
NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)
More informationZARISKI-LIKE TOPOLOGY ON THE CLASSICAL PRIME SPECTRUM OF A MODULE
Bulletin of the Iranian Mathematical Society Vol. 35 No. 1 (2009), pp 253-269. ZARISKI-LIKE TOPOLOGY ON THE CLASSICAL PRIME SPECTRUM OF A MODULE M. BEHBOODI AND M. J. NOORI Abstract. Let R be a commutative
More informationarxiv: v1 [math.ac] 25 Jul 2017
Primary Decomposition in Boolean Rings David C. Vella, Skidmore College arxiv:1707.07783v1 [math.ac] 25 Jul 2017 1. Introduction Let R be a commutative ring with identity. The Lasker- Noether theorem on
More informationA NEW LINDELOF SPACE WITH POINTS G δ
A NEW LINDELOF SPACE WITH POINTS G δ ALAN DOW Abstract. We prove that implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 ℵ1 which has points G δ. In addition, this space has
More informationMinimal Prime Element Space of an Algebraic Frame. Papiya Bhattacharjee. A Dissertation
Minimal Prime Element Space of an Algebraic Frame Papiya Bhattacharjee A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for
More informationALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!
ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.
More informationA Note on the Inverse Limits of Linear Algebraic Groups
International Journal of Algebra, Vol. 5, 2011, no. 19, 925-933 A Note on the Inverse Limits of Linear Algebraic Groups Nadine J. Ghandour Math Department Lebanese University Nabatieh, Lebanon nadine.ghandour@liu.edu.lb
More informationON MAXIMAL, MINIMAL OPEN AND CLOSED SETS
Commun. Korean Math. Soc. 30 (2015), No. 3, pp. 277 282 http://dx.doi.org/10.4134/ckms.2015.30.3.277 ON MAXIMAL, MINIMAL OPEN AND CLOSED SETS Ajoy Mukharjee Abstract. We obtain some conditions for disconnectedness
More informationON RESOLVABLE PRIMAL SPACES
The Version of Record of this manuscript has been published and is available in Quaestiones Mathematicae (2018) https://doiorg/102989/1607360620181437093 ON RESOLVABLE PRIMAL SPACES Intissar Dahane Department
More informationLOCALLY PRINCIPAL IDEALS AND FINITE CHARACTER arxiv: v1 [math.ac] 16 May 2013
LOCALLY PRINCIPAL IDEALS AND FINITE CHARACTER arxiv:1305.3829v1 [math.ac] 16 May 2013 STEFANIA GABELLI Abstract. It is well-known that if R is a domain with finite character, each locally principal nonzero
More informationThe Zariski topology on the set of semistar operations on an integral domain
Università degli Studi Roma Tre Dipartimento di Matematica e Fisica Corso di Laurea Magistrale in Matematica Tesi di Laurea Magistrale in Matematica Master-level Thesis The Zariski topology on the set
More informationThis chapter contains a very bare summary of some basic facts from topology.
Chapter 2 Topological Spaces This chapter contains a very bare summary of some basic facts from topology. 2.1 Definition of Topology A topology O on a set X is a collection of subsets of X satisfying the
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationOUTER MEASURES ON A COMMUTATIVE RING INDUCED BY MEASURES ON ITS SPECTRUM. Dariusz Dudzik, Marcin Skrzyński. 1. Preliminaries and introduction
Annales Mathematicae Silesianae 31 (2017), 63 70 DOI: 10.1515/amsil-2016-0020 OUTER MEASURES ON A COMMUTATIVE RING INDUCED BY MEASURES ON ITS SPECTRUM Dariusz Dudzik, Marcin Skrzyński Abstract. On a commutative
More informationRestricted versions of the Tukey-Teichmüller Theorem that are equivalent to the Boolean Prime Ideal Theorem
Restricted versions of the Tukey-Teichmüller Theorem that are equivalent to the Boolean Prime Ideal Theorem R.E. Hodel Dedicated to W.W. Comfort on the occasion of his seventieth birthday. Abstract We
More informationP-Ideals and PMP-Ideals in Commutative Rings
Journal of Mathematical Extension Vol. 10, No. 4, (2016), 19-33 Journal ISSN: 1735-8299 of Mathematical Extension Vol. URL: 10, http://www.ijmex.com No. 4, (2016), 19-33 ISSN: 1735-8299 URL: http://www.ijmex.com
More informations P = f(ξ n )(x i x i 1 ). i=1
Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological
More informationTopology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng
Topology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng Remark 0.1. This is a solution Manuel to the topology questions of the Topology Geometry
More informationPREOPEN SETS AND RESOLVABLE SPACES
PREOPEN SETS AND RESOLVABLE SPACES Maximilian Ganster appeared in: Kyungpook Math. J. 27 (2) (1987), 135 143. Abstract This paper presents solutions to some recent questions raised by Katetov about the
More informationA CHARACTERIZATION OF POWER HOMOGENEITY G. J. RIDDERBOS
A CHARACTERIZATION OF POWER HOMOGENEITY G. J. RIDDERBOS Abstract. We prove that every -power homogeneous space is power homogeneous. This answers a question of the author and it provides a characterization
More informationarxiv: v1 [math.ra] 11 Jul 2018
On the prime spectrum of an le-module arxiv:1807.04024v1 [math.ra] 11 Jul 2018 M. Kumbhakar and A. K. Bhuniya Department of Mathematics, Nistarini College, Purulia-723101, W.B. Department of Mathematics,
More informationOn z -ideals in C(X) F. A z a r p a n a h, O. A. S. K a r a m z a d e h and A. R e z a i A l i a b a d (Ahvaz)
F U N D A M E N T A MATHEMATICAE 160 (1999) On z -ideals in C(X) by F. A z a r p a n a h, O. A. S. K a r a m z a d e h and A. R e z a i A l i a b a d (Ahvaz) Abstract. An ideal I in a commutative ring
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationABSTRACT NONSINGULAR CURVES
ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a
More informationRELATIONS BETWEEN UNION AND INTERSECTION OF IDEALS AND THEIR CORRESPONDING IDEAL TOPOLOGIES 1
Novi Sad J. Math. Vol. 45, No. 2, 2015, 39-46 RELATIONS BETWEEN UNION AND INTERSECTION OF IDEALS AND THEIR CORRESPONDING IDEAL TOPOLOGIES 1 S. Suriyakala 2 and R. Vembu 3 Abstract. The concept of ideals
More informationZeta Functions of Burnside Rings for Symmetric and Alternating Groups
International Journal of Algebra, Vol. 6, 2012, no. 25, 1207-1220 Zeta Functions of Burnside Rings for Symmetric and Alternating Groups David Villa-Hernández Benemérita Universidad Autónoma de Puebla Facultad
More informationMAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.
MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar
More informationA NOTE ON GOLOMB TOPOLOGIES
A NOTE ON GOLOMB TOPOLOGIES PETE L. CLARK, NOAH LEBOWITZ-LOCKARD, AND PAUL POLLACK Abstract. In 1959 Golomb defined a connected topology on Z. An analogous Golomb topology on an arbitrary integral domain
More informationOn Strongly Prime Semiring
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 30(2) (2007), 135 141 On Strongly Prime Semiring T.K. Dutta and M.L. Das Department
More informationThe Zariski Spectrum of a ring
Thierry Coquand September 2010 Use of prime ideals Let R be a ring. We say that a 0,..., a n is unimodular iff a 0,..., a n = 1 We say that Σa i X i is primitive iff a 0,..., a n is unimodular Theorem:
More informationTHE IMPLICIT FUNCTION THEOREM FOR CONTINUOUS FUNCTIONS. Carlos Biasi Carlos Gutierrez Edivaldo L. dos Santos. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 32, 2008, 177 185 THE IMPLICIT FUNCTION THEOREM FOR CONTINUOUS FUNCTIONS Carlos Biasi Carlos Gutierrez Edivaldo L.
More informationarxiv: v1 [math.ds] 22 Jan 2019
A STUDY OF HOLOMORPHIC SEMIGROUPS arxiv:1901.07364v1 [math.ds] 22 Jan 2019 BISHNU HARI SUBEDI Abstract. In this paper, we investigate some characteristic features of holomorphic semigroups. In particular,
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationObstinate filters in residuated lattices
Bull. Math. Soc. Sci. Math. Roumanie Tome 55(103) No. 4, 2012, 413 422 Obstinate filters in residuated lattices by Arsham Borumand Saeid and Manijeh Pourkhatoun Abstract In this paper we introduce the
More informationTROPICAL SCHEME THEORY. Idempotent semirings
TROPICAL SCHEME THEORY Idempotent semirings Definition 0.1. A semiring is (R, +,, 0, 1) such that (R, +, 0) is a commutative monoid (so + is a commutative associative binary operation on R 0 is an additive
More informationExtended Index. 89f depth (of a prime ideal) 121f Artin-Rees Lemma. 107f descending chain condition 74f Artinian module
Extended Index cokernel 19f for Atiyah and MacDonald's Introduction to Commutative Algebra colon operator 8f Key: comaximal ideals 7f - listings ending in f give the page where the term is defined commutative
More informationPrime and Irreducible Ideals in Subtraction Algebras
International Mathematical Forum, 3, 2008, no. 10, 457-462 Prime and Irreducible Ideals in Subtraction Algebras Young Bae Jun Department of Mathematics Education Gyeongsang National University, Chinju
More informationCOMPACT ORTHOALGEBRAS
COMPACT ORTHOALGEBRAS ALEXANDER WILCE Abstract. We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and
More informationA GENERALIZATION OF BI IDEALS IN SEMIRINGS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 8(2018), 123-133 DOI: 10.7251/BIMVI1801123M Former BULLETIN
More informationCommutative Algebra. Timothy J. Ford
Commutative Algebra Timothy J. Ford DEPARTMENT OF MATHEMATICS, FLORIDA ATLANTIC UNIVERSITY, BOCA RA- TON, FL 33431 Email address: ford@fau.edu URL: http://math.fau.edu/ford Last modified January 9, 2018.
More information2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α.
Chapter 2. Basic Topology. 2.3 Compact Sets. 2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α. 2.32 Definition A subset
More informationDistributivity of Quotients of Countable Products of Boolean Algebras
Rend. Istit. Mat. Univ. Trieste Volume 41 (2009), 27 33. Distributivity of Quotients of Countable Products of Boolean Algebras Fernando Hernández-Hernández Abstract. We compute the distributivity numbers
More informationSTRONGLY CONNECTED SPACES
Undergraduate Research Opportunity Programme in Science STRONGLY CONNECTED SPACES Submitted by Dai Bo Supervised by Dr. Wong Yan-loi Department of Mathematics National University of Singapore Academic
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More informationSpectral Compactification of a Ring
International Mathematical Forum, Vol. 7, 2012, no. 19, 925-934 Spectral Compactification of a Ring Lorenzo Acosta G. Departamento de Matemáticas Universidad Nacional de Colombia Sede Bogotá, Colombia
More informationFinite groups determined by an inequality of the orders of their elements
Publ. Math. Debrecen 80/3-4 (2012), 457 463 DOI: 10.5486/PMD.2012.5168 Finite groups determined by an inequality of the orders of their elements By MARIUS TĂRNĂUCEANU (Iaşi) Abstract. In this note we introduce
More informationOn poset Boolean algebras
1 On poset Boolean algebras Uri Abraham Department of Mathematics, Ben Gurion University, Beer-Sheva, Israel Robert Bonnet Laboratoire de Mathématiques, Université de Savoie, Le Bourget-du-Lac, France
More informationINVARIANT PROBABILITIES ON PROJECTIVE SPACES. 1. Introduction
INVARIANT PROBABILITIES ON PROJECTIVE SPACES YVES DE CORNULIER Abstract. Let K be a local field. We classify the linear groups G GL(V ) that preserve an probability on the Borel subsets of the projective
More informationPrime k-bi-ideals in Γ-Semirings
Palestine Journal of Mathematics Vol. 3(Spec 1) (2014), 489 494 Palestine Polytechnic University-PPU 2014 Prime k-bi-ideals in Γ-Semirings R.D. Jagatap Dedicated to Patrick Smith and John Clark on the
More informationON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET
Novi Sad J. Math. Vol. 40, No. 2, 2010, 7-16 ON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET M.K. Bose 1, Ajoy Mukharjee 2 Abstract Considering a countable number of topologies on a set X, we introduce the
More informationSUBCATEGORIES OF EXTENSION MODULES BY SERRE SUBCATEGORIES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 7, July 2012, Pages 2293 2305 S 0002-9939(2011)11108-0 Article electronically published on November 23, 2011 SUBCATEGORIES OF EXTENSION
More informationMath 418 Algebraic Geometry Notes
Math 418 Algebraic Geometry Notes 1 Affine Schemes Let R be a commutative ring with 1. Definition 1.1. The prime spectrum of R, denoted Spec(R), is the set of prime ideals of the ring R. Spec(R) = {P R
More informationDenition.9. Let a A; t 0; 1]. Then by a fuzzy point a t we mean the fuzzy subset of A given below: a t (x) = t if x = a 0 otherwise Denition.101]. A f
Some Properties of F -Spectrum of a Bounded Implicative BCK-Algebra A.Hasankhani Department of Mathematics, Faculty of Mathematical Sciences, Sistan and Baluchestan University, Zahedan, Iran Email:abhasan@hamoon.usb.ac.ir,
More informationThe Zariski topology on sets of semistar operations
The Zariski topology on sets of semistar operations Dario Spirito (joint work with Carmelo Finocchiaro and Marco Fontana) Università di Roma Tre February 13th, 2015 Dario Spirito (Univ. Roma Tre) Topology
More informationAlgebraic properties of rings of continuous functions
F U N D A M E N T A MATHEMATICAE 149 (1996) Algebraic properties of rings of continuous functions by M. A. M u l e r o (Badajoz) Abstract. This paper is devoted to the study of algebraic properties of
More informationDENSELY k-separable COMPACTA ARE DENSELY SEPARABLE
DENSELY k-separable COMPACTA ARE DENSELY SEPARABLE ALAN DOW AND ISTVÁN JUHÁSZ Abstract. A space has σ-compact tightness if the closures of σ-compact subsets determines the topology. We consider a dense
More informationZARISKI TOPOLOGY FOR SECOND SUBHYPERMODULES
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 2018 (554 568) 554 ZARISKI TOPOLOGY FOR SECOND SUBHYPERMODULES Razieh Mahjoob Vahid Ghaffari Department of Mathematics Faculty of Mathematics Statistics
More informationSplitting sets and weakly Matlis domains
Commutative Algebra and Applications, 1 8 de Gruyter 2009 Splitting sets and weakly Matlis domains D. D. Anderson and Muhammad Zafrullah Abstract. An integral domain D is weakly Matlis if the intersection
More informationMV-algebras and fuzzy topologies: Stone duality extended
MV-algebras and fuzzy topologies: Stone duality extended Dipartimento di Matematica Università di Salerno, Italy Algebra and Coalgebra meet Proof Theory Universität Bern April 27 29, 2011 Outline 1 MV-algebras
More informationTroisième Rencontre Internationale sur les Polynômes à Valeurs Entières
Troisième Rencontre Internationale sur les Polynômes à Valeurs Entières Rencontre organisée par : Sabine Evrard 29 novembre-3 décembre 2010 Carmelo Antonio Finocchiaro and Marco Fontana Some applications
More information3 Lecture 3: Spectral spaces and constructible sets
3 Lecture 3: Spectral spaces and constructible sets 3.1 Introduction We want to analyze quasi-compactness properties of the valuation spectrum of a commutative ring, and to do so a digression on constructible
More informationProperties of the Integers
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationSubdirectly irreducible commutative idempotent semirings
Subdirectly irreducible commutative idempotent semirings Ivan Chajda Helmut Länger Palacký University Olomouc, Olomouc, Czech Republic, email: ivan.chajda@upol.cz Vienna University of Technology, Vienna,
More informationMAPPING CHAINABLE CONTINUA ONTO DENDROIDS
MAPPING CHAINABLE CONTINUA ONTO DENDROIDS PIOTR MINC Abstract. We prove that every chainable continuum can be mapped into a dendroid such that all point-inverses consist of at most three points. In particular,
More informationa + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationTopological groups with dense compactly generated subgroups
Applied General Topology c Universidad Politécnica de Valencia Volume 3, No. 1, 2002 pp. 85 89 Topological groups with dense compactly generated subgroups Hiroshi Fujita and Dmitri Shakhmatov Abstract.
More informationPRIME RADICAL IN TERNARY HEMIRINGS. R.D. Giri 1, B.R. Chide 2. Shri Ramdeobaba College of Engineering and Management Nagpur, , INDIA
International Journal of Pure and Applied Mathematics Volume 94 No. 5 2014, 631-647 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v94i5.1
More informationCOINCIDENCE AND THE COLOURING OF MAPS
COINCIDENCE AND THE COLOURING OF MAPS JAN M. AARTS AND ROBBERT J. FOKKINK ABSTRACT In [8, 6] it was shown that for each k and n such that 2k n, there exists a contractible k-dimensional complex Y and a
More informationZERO-DIMENSIONALITY AND SERRE RINGS. D. Karim
Serdica Math. J. 30 (2004), 87 94 ZERO-DIMENSIONALITY AND SERRE RINGS D. Karim Communicated by L. Avramov Abstract. This paper deals with zero-dimensionality. We investigate the problem of whether a Serre
More informationOn A Weaker Form Of Complete Irresoluteness. Key Words: irresolute function, δ-semiopen set, regular open set. Contents.
Bol. Soc. Paran. Mat. (3s.) v. 26 1-2 (2008): 81 87. c SPM ISNN-00378712 On A Weaker Form Of Complete Irresoluteness Erdal Ekici and Saeid Jafari abstract: The aim of this paper is to present a new class
More informationCONNECTEDNESS IN IDEAL TOPOLOGICAL SPACES
Novi Sad J. Math. Vol. 38, No. 2, 2008, 65-70 CONNECTEDNESS IN IDEAL TOPOLOGICAL SPACES Erdal Ekici 1, Takashi Noiri 2 Abstract. In this paper we study the notion of connectedness in ideal topological
More informationIntroduction to Dynamical Systems
Introduction to Dynamical Systems France-Kosovo Undergraduate Research School of Mathematics March 2017 This introduction to dynamical systems was a course given at the march 2017 edition of the France
More informationOn Almost Continuity and Expansion of Open Sets
On Almost Continuity and Expansion of Open Sets Sobre Casi Continuidad y Expansión de Conjuntos Abiertos María Luisa Colasante (marucola@ciens.ula.ve) Departamento de Matemáticas Facultad de Ciencias,
More informationThe Space of Minimal Prime Ideals of C(x) Need not be Basically Disconnected
Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 9-1-1988 The Space of Minimal Prime Ideals of C(x) Need not be Basically Disconnected Alan Dow
More informationCHAPTER 1. AFFINE ALGEBRAIC VARIETIES
CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the
More informationATOMIC AND AP SEMIGROUP RINGS F [X; M], WHERE M IS A SUBMONOID OF THE ADDITIVE MONOID OF NONNEGATIVE RATIONAL NUMBERS. Ryan Gipson and Hamid Kulosman
International Electronic Journal of Algebra Volume 22 (2017) 133-146 DOI: 10.24330/ieja.325939 ATOMIC AND AP SEMIGROUP RINGS F [X; M], WHERE M IS A SUBMONOID OF THE ADDITIVE MONOID OF NONNEGATIVE RATIONAL
More informationTHE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES
THE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES JULIO BECERRA GUERRERO AND ANGEL RODRIGUEZ PALACIOS 1. Introduction Throughout this paper, X will denote a Banach space, S S(X) and B B(X) will be the unit
More informationSets of Lengths of Puiseux Monoids
UC Berkeley Conference on Rings and Factorizations Institute of Mathematics and Scientific Computing University of Graz, Austria February 21, 2018 Introduction Online reference: https://arxiv.org/abs/1711.06961
More information