Lecture 22 More on Rings
Last time: A ring is a set R with two binary operations, and, such that pr, q is an abelian group, pr, q is a semigroup (no identity or inverses), and the distributive property holds.
Last time: A ring is a set R with two binary operations, and, such that pr, q is an abelian group, pr, q is a semigroup (no identity or inverses), and the distributive property holds. A zero divisor is an element a P R such that ab 0 or ba 0 for some b. A unit is an element of a ring with 1 that has an inverse.
Last time: A ring is a set R with two binary operations, and, such that pr, q is an abelian group, pr, q is a semigroup (no identity or inverses), and the distributive property holds. A zero divisor is an element a P R such that ab 0 or ba 0 for some b. A unit is an element of a ring with 1 that has an inverse. A ring is commutative if pr, q is commutative. (e.g. Z) An integral domain is where R has no zero divisors. (e.g. Z) (Recall: I.D. s have cancellation properties for non-zero-divisors.) A division ring is where pr, q is a group. (e.g. Q) A field is where pr, q is an abelian group. (e.g. Q)
More examples, more formality: Polynomial rings Definition Let R be a commutative ring with identity. The formal sum a n x n a n 1 x n 1 a 1 x a 0 with n 0 and each a i P R is called a polynomial in x with coefficients a i in R.
More examples, more formality: Polynomial rings Definition Let R be a commutative ring with identity. The formal sum a n x n a n 1 x n 1 a 1 x a 0 with n 0 and each a i P R is called a polynomial in x with coefficients a i in R. If a n 0, the polynomial is of degree n, a n x n is the leading term and a n is the leading coefficient. The polynomial is monic if a n 1.
More examples, more formality: Polynomial rings Definition Let R be a commutative ring with identity. The formal sum a n x n a n 1 x n 1 a 1 x a 0 with n 0 and each a i P R is called a polynomial in x with coefficients a i in R. If a n 0, the polynomial is of degree n, a n x n is the leading term and a n is the leading coefficient. The polynomial is monic if a n 1. The set of all such polynomials is the ring of polynomials in the x with coefficients in R, denoted Rrxs. The ring R appears in Rrxs as the constant polynomials.
More examples, more formality: Polynomial rings If R is not an integral domain, then neither is Rrxs.
More examples, more formality: Polynomial rings If R is not an integral domain, then neither is Rrxs. If S R is a subring, then Srxs Rrxs is a subring.
More examples, more formality: Polynomial rings If R is not an integral domain, then neither is Rrxs. If S R is a subring, then Srxs Rrxs is a subring. Proposition Let R be an integral domain and let ppxq, qpxq P Rrxszt0u. Then 1. degree ppxqqpxq degree ppxq degree qpxq, 2. the units of Rrxs are the units of R, 3. Rrxs is an integral domain.
More examples, more formality: Other rings with variables Let R be a ring with 1. The field of fractions of Rrxs is tppxq{qpxq ppxq, qpxq P Rrxs, qpxq 0u.
More examples, more formality: Other rings with variables Let R be a ring with 1. The field of fractions of Rrxs is tppxq{qpxq ppxq, qpxq P Rrxs, qpxq 0u. The field of fractions over a field F is written F pxq. The field of fractions of Zrxs is Qpxq.
More examples, more formality: Other rings with variables Let R be a ring with 1. The field of fractions of Rrxs is tppxq{qpxq ppxq, qpxq P Rrxs, qpxq 0u. The field of fractions over a field F is written F pxq. The field of fractions of Zrxs is Qpxq. The ring of formal power series over R is Rrrxss # 8 n0 a n x n a n P R ( formal means need not deal with convergence) +.
More examples, more formality: Other rings with variables Let R be a ring with 1. The field of fractions of Rrxs is tppxq{qpxq ppxq, qpxq P Rrxs, qpxq 0u. The field of fractions over a field F is written F pxq. The field of fractions of Zrxs is Qpxq. The ring of formal power series over R is Rrrxss # 8 n0 a n x n a n P R ( formal means need not deal with convergence) If F is a field, the field of fractions over F rrxss is F ppxqq # n N + a n x n a n P F, N P Z. +.
More examples, more formality: Matrix rings Definition Let R be a ring and n a positive integer. Let M n prq be the set of all n n matrices with entries from R.
More examples, more formality: Matrix rings Definition Let R be a ring and n a positive integer. Let M n prq be the set of all n n matrices with entries from R. Facts: The set M n prq forms a ring under addition and multiplication. It is not commutative, and it is not a division ring.
More examples, more formality: Matrix rings Definition Let R be a ring and n a positive integer. Let M n prq be the set of all n n matrices with entries from R. Facts: The set M n prq forms a ring under addition and multiplication. It is not commutative, and it is not a division ring. The scalar matrices are those diagonal matrices of the form a id with a P R. They form a subring of M n prq isomorphic to R (we ll define isomorphic later)
More examples, more formality: Matrix rings Definition Let R be a ring and n a positive integer. Let M n prq be the set of all n n matrices with entries from R. Facts: The set M n prq forms a ring under addition and multiplication. It is not commutative, and it is not a division ring. The scalar matrices are those diagonal matrices of the form a id with a P R. They form a subring of M n prq isomorphic to R (we ll define isomorphic later) Other subrings: upper and lower triangular matrices, and if S R, then M n psq.
More examples, more formality: Group rings Fix a ring R with identity and a finite group G. Let RG # gpg be the group ring of G over R. a g g a g P R, g P G +
More examples, more formality: Group rings Fix a ring R with identity and a finite group G. Let RG # gpg a g g a g P R, g P G be the group ring of G over R. With addition component-wise: if a gpg a gg and b gpg b gg, then a b gpg pa g b g qg +
More examples, more formality: Group rings Fix a ring R with identity and a finite group G. Let RG # gpg a g g a g P R, g P G be the group ring of G over R. With addition component-wise: if a gpg a gg and b gpg b gg, then a b gpg and multiplication like polynomials: pa g b g qg pa g gqpb b hq pa g b h qpghq and expansion according to distributive laws. +