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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 an xn with n ¥ 0 and each ai coefficients ai in R. an1 xn1 a1 x a0 P R is called a polynomial in x with More examples, more formality: Polynomial rings Definition Let R be a commutative ring with identity. The formal sum an xn an1 xn1 a1 x a0 with n ¥ 0 and each ai P R is called a polynomial in x with coefficients ai in R. If an 0, the polynomial is of degree n, an xn is the leading term and an is the leading coefficient. The polynomial is monic if an 1. More examples, more formality: Polynomial rings Definition Let R be a commutative ring with identity. The formal sum an xn an1 xn1 a1 x a0 with n ¥ 0 and each ai P R is called a polynomial in x with coefficients ai in R. If an 0, the polynomial is of degree n, an xn is the leading term and an is the leading coefficient. The polynomial is monic if an 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 S rxs ¤ 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 S rxs ¤ Rrxs is a subring. Proposition Let R be an integral domain and let ppxq, q pxq P Rrxszt0u. Then 1. degree ppxqq pxq degree ppxq degree q pxq, 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 # 8̧ an x n | an P R + . n 0 (“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 # 8̧ an x n | an P R + . n 0 (“formal” means need not deal with convergence) If F is a field, the field of fractions over F rrxss is F ppxqq #¸ ¥ n N an xn | an + P F, N P Z . More examples, more formality: Matrix rings Definition Let R be a ring and n a positive integer. Let Mn 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 Mn pRq be the set of all n n matrices with entries from R. Facts: The set Mn 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 Mn pRq be the set of all n n matrices with entries from R. Facts: The set Mn 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 Mn 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 Mn pRq be the set of all n n matrices with entries from R. Facts: The set Mn 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 Mn pRq isomorphic to R (we’ll define isomorphic later) Other subrings: upper and lower triangular matrices, and if S then Mn pS q. ¤ R, More examples, more formality: Group rings Fix a ring R with identity and a finite group G. Let # RG ¸ P ag g | ag g G be the group ring of G over R. P R, g P G + More examples, more formality: Group rings Fix a ring R with identity and a finite group G. Let # RG ¸ P ag g | ag P R, g P G g G be the group ring of G over R. With addition component-wise: ° ° if a gPG ag g and b gPG bg g, then a b ¸ P g G pa g bg qg + More examples, more formality: Group rings Fix a ring R with identity and a finite group G. Let # RG ¸ P ag g | ag P R, g P G g G be the group ring of G over R. With addition component-wise: ° ° if a gPG ag g and b gPG bg g, then a b ¸ P pa g bg qg g G and multiplication like polynomials: pag gqpbbhq pag bhqpghq and expansion according to distributive laws. +