Lecture 24 Properties of deals

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1 Lecture 24 Properties of deals

2 Aside: Representation theory of finite groups Let G be a finite group, and let R C, R, or Q (any commutative ring).

3 Aside: Representation theory of finite groups Let G be a finite group, and let R C, R, or Q (any commutative ring). Suppose G acts on the set R n in such a way that the action just on the elements of a basis linearly extends to an action on the whole vector space.

4 Aside: Representation theory of finite groups Let G be a finite group, and let R C, R, or Q (any commutative ring). Suppose G acts on the set R n in such a way that the action just on the elements of a basis linearly extends to an action on the whole vector space. Example: Z 2 xxy acts on R 2 generated by x p1, 0q p0, 1q and x p0, 1q p1, 0q

5 Aside: Representation theory of finite groups Let G be a finite group, and let R C, R, or Q (any commutative ring). Suppose G acts on the set R n in such a way that the action just on the elements of a basis linearly extends to an action on the whole vector space. Example: Z 2 xxy acts on R 2 generated by x p1, 0q p0, 1q and x p0, 1q p1, 0q To generate the whole action, extend linearly, i.e. x pa, bq x pap1, 0q apx p1, 0qq ap0, 1q bp0, 1qq bpx p0, 1qq bp1, 0q pb, aq

6 Aside: Representation theory of finite groups Let G be a finite group, and let R C, R, or Q (any commutative ring). Suppose G acts on the set R n in such a way that the action just on the elements of a basis linearly extends to an action on the whole vector space. Example: Z 2 xxy acts on R 2 generated by x p1, 0q p0, 1q and x p0, 1q p1, 0q To generate the whole action, extend linearly, i.e. x pa, bq x pap1, 0q apx p1, 0qq ap0, 1q bp0, 1qq bpx p0, 1qq bp1, 0q pb, aq Then G s action extends to an action of the group algebra RG!

7 Aside: Representation theory of finite groups Let G be a finite group, and let R C, R, or Q (any commutative ring). Suppose G acts on the set R n in such a way that the action just on the elements of a basis linearly extends to an action on the whole vector space. Example: Z 2 xxy acts on R 2 generated by x p1, 0q p0, 1q and x p0, 1q p1, 0q To generate the whole action, extend linearly, i.e. x pa, bq x pap1, 0q apx p1, 0qq ap0, 1q bp0, 1qq bpx p0, 1qq bp1, 0q pb, aq Then G s action extends to an action of the group algebra RG! Formally, a ring S acts on an abelian additive group A if 0 a 0, 1 a a, s pa bq s a s b, ps s 1 q a ps aq ps 1 aq, pss 1 q a s ps 1 aq for all a, b P A, s, s 1 P s. (A is called an S-module)

8 Last time: A ring homomorphism is a map ϕ : R Ñ S such that for all a, b P R ϕpa bq ϕpaq ϕpbq and ϕpabq ϕpaqϕpbq. The kernel of the ring homomorphism ϕ is the the kernel of the underlying group homomorphism, i.e. kerϕ ta P R ϕpaq 0 P Su.

9 Last time: A ring homomorphism is a map ϕ : R Ñ S such that for all a, b P R ϕpa bq ϕpaq ϕpbq and ϕpabq ϕpaqϕpbq. The kernel of the ring homomorphism ϕ is the the kernel of the underlying group homomorphism, i.e. kerϕ ta P R ϕpaq 0 P Su. Both kerϕ and imgϕ are rings, as is the group R{kerϕ with multiplication pr 1 kerϕqpr 2 kerϕq r 1 r 2 kerϕ.

10 Last time: In general: for any subset A R, define the set R{A tr A r P Ru, where r A tr a a P Au. When A is a group, we get all of our coset knowledge back, like r A s A if and only if r s P A.

11 Last time: In general: for any subset A R, define the set R{A tr A r P Ru, where r A tr a a P Au. When A is a group, we get all of our coset knowledge back, like r A s A if and only if r s P A. Big question: For which subsets A R is R{A a well-defined ring with operations pr Aq pr 1 Aq pr r 1 q A, and (1) pr Aqpr 1 Aq rr 1 A? (2)

12 Last time: In general: for any subset A R, define the set R{A tr A r P Ru, where r A tr a a P Au. When A is a group, we get all of our coset knowledge back, like r A s A if and only if r s P A. Big question: For which subsets A R is R{A a well-defined ring with operations pr Aq pr 1 Aq pr r 1 q A, and (1) pr Aqpr 1 Aq rr 1 A? (2) ( ) From Ch 3, we know (1) forces A pr, q (Any subgroup will do because pr, q is abelian)

13 Last time: In general: for any subset A R, define the set R{A tr A r P Ru, where r A tr a a P Au. When A is a group, we get all of our coset knowledge back, like r A s A if and only if r s P A. Big question: For which subsets A R is R{A a well-defined ring with operations pr Aq pr 1 Aq pr r 1 q A, and (1) pr Aqpr 1 Aq rr 1 A? (2) ( ) From Ch 3, we know (1) forces A pr, q (Any subgroup will do because pr, q is abelian) ( ) For (2), consider the cases r r 1 0 forces A is a subring of R r 0 forces Ar 1 A (A is a right ideal) r 1 0 forces ra A (A is a left ideal)

14 Quotient rings Recall, an ideal of a ring R is a subring I R which satisfies ri I and Ir I for all r P R.

15 Quotient rings Recall, an ideal of a ring R is a subring I R which satisfies ri I and Ir I for all r P R. Proposition Let R be a ring and I an ideal of R. Then the quotient group R{I is a ring via pr Iq ps Iq pr s Iq and pr Iq ps Iq prs Iq for all r, s P R. Conversely, if I is a subgroup of R such that the above operations are well defined, then I is an ideal of R.

16 Quotient rings Recall, an ideal of a ring R is a subring I R which satisfies ri I and Ir I for all r P R. Proposition Let R be a ring and I an ideal of R. Then the quotient group R{I is a ring via pr Iq ps Iq pr s Iq and pr Iq ps Iq prs Iq for all r, s P R. Conversely, if I is a subgroup of R such that the above operations are well defined, then I is an ideal of R. Definition When I is an ideal of R, the ring R{I with the operations above is called the quotient ring of R by I.

17 Isomorphism theorems Theorem 1. (First iso thm for rings) If ϕ : R Ñ S is a homomorphism of rings, then kerϕ is an ideal of R, the image of ϕ is a subring of S and R{kerϕ is isomorphic as a ring to ϕprq.

18 Isomorphism theorems Theorem 1. (First iso thm for rings) If ϕ : R Ñ S is a homomorphism of rings, then kerϕ is an ideal of R, the image of ϕ is a subring of S and R{kerϕ is isomorphic as a ring to ϕprq. 2. If I is an ideal of R, then the natural projection R Ñ R{I defined by r ÞÑ r I is a surjective ring homomorphism with kernel I. Thus every ideal is the kernel of some ring homomorphism and vice versa.

19 More isomorphism theorems Theorem Let R be a ring. 1. (Second isomorphism theorem for rings) Let A be a subring and B be an ideal of R. Then A B ta b : a P A, b P Bu is a subring of R, A X B is an ideal of A and pa Bq{B A{pA X Bq. 2. (Third isomorphism theorem for rings) Let I and J be ideals of R with I J. Then J{I is an ideal of R{I and pr{iq{pj{iq R{J. 3. (Fourth isomorphism theorem for ring) Let I be an ideal of R. The correspondence A Ø A{I is an inclusion preserving bijection between the subrings A of R that contain I and the set of subrings of R{I. Furthermore, A is an ideal of R if and only if A{I is an ideal of R{I.

20 Building ideals from ideals Definition Let I and J be ideals of R. 1. Define the sum of I and J to be I J ti j : i P I, j P Ju. (fact: the smallest ideal containing both I and J)

21 Building ideals from ideals Definition Let I and J be ideals of R. 1. Define the sum of I and J to be I J ti j : i P I, j P Ju. (fact: the smallest ideal containing both I and J) 2. Define the product of I and J to be IJ t finite sums of elements of the form ij i P I, j P Ju. (fact: IJ is an ideal contained in I X J)

22 Building ideals from ideals Definition Let I and J be ideals of R. 1. Define the sum of I and J to be I J ti j : i P I, j P Ju. (fact: the smallest ideal containing both I and J) 2. Define the product of I and J to be IJ t finite sums of elements of the form ij i P I, j P Ju. 3. For any n 1, define the nth power of I (fact: IJ is an ideal contained in I X J) I n t finite sums of elements of the form a 1... a n a i P Iu

23 Building ideals from ideals Definition Let I and J be ideals of R. 1. Define the sum of I and J to be I J ti j : i P I, j P Ju. (fact: the smallest ideal containing both I and J) 2. Define the product of I and J to be IJ t finite sums of elements of the form ij i P I, j P Ju. 3. For any n 1, define the nth power of I (fact: IJ is an ideal contained in I X J) I n t finite sums of elements of the form a 1... a n a i P Iu In summary, if I and J are ideals, then I X J, I J, and IJ

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