7.2 EXAMPLES: POLYNOMIAL RINGS, MATRIX RINGS AND GROUP RINGS Definition. Let B be a set and ∅ 6= S ⊆ B × B. A partial product ? on B (with domain S) is a map ? : S → B :(i, j) 7→ i ? j which is associative in the following sense: Whenever one side of (i ? j) ? k = i ? (j ? k) is defined, the other one is also, and the two sides evaluate to the same element of B. It is also assumed that for every k ∈ B the set {(i, j) ∈ S | i ? j = k} is finite. Examples. 1. S = diag(B) := {(i, i) ∈ B × B} and i ? i = i . 2. B = N = {0, 1, 2,...} S = N × N and i ? j := i + j . 3. n ∈ Z+ n = {0, 1, 2, . , n} B = n × n S = {((i, j), (j, k))} and (i, j) ? (j, k) = (i, k) . 4. B = G, S = G × G and i ? j := ij (the product in G). Definition Given a ring R and a partial product ? : S → B :(i, j) 7→ i ? j on a set B let RB denote the abelian group of all functions from B to R with pointwise addition. Define a product for RB as follows: X If f ∈ RB, g ∈ RB set (fg)(k) := f(i)g(j) . S3(i,j) | i?j=k Exercise Verify that this definition yields a ring structure with identity on RB . Continuing with the examples of partial products enumerated above one obtains the corresponding rings: 1. RB is the ring of functions from B to R with pointwise operations. 2. If R is commutative and has 1 ∈ R then RN is the ring of formal power series with coefficients in R also denoted by R[[x]] . The usual notation equates functions and formal series as follows ∞ X RN 3 f : k 7→ f(k) ⇔ f(k)xk ∈ R[[x]] . k=0 3. The ring Rn×n is the ring of n × n matrices with entries from R, which is usually denoted by Mn(R) . 4. If R is commutative and has 1 ∈ R then the ring RG, where G is a group, is the group ring of G with coefficients in R, which is usually denoted by RG . Polynomial rings. Given the ring of formal power series in x RN = R[[x]] define the subring of power series in x as ∞ ( ) X k R N = R[x] := {f = f(k)x | f(k) = 0 for almost all k ∈ N} . k=0 The elements of this ring are called polynomials in x. The zero polynomial 0R[x] corresponds to the function (k 7→ 0) . By definition it has no degree. For a polynomial f 6= 0 its degree is defined as deg(f) := max{k ∈ N | f(k) 6= 0} ∈ N . Polynomials of degree 1 are called constant polynomials. Polynomials of degree 2 are called linear polynomials. R[x] is a commutative ring with identity which contains a copy of R as a subring via R 3 a 7→ a1R[x] ∈ R[x] . Proposition 4 Let R be an integral domain and let p, q ∈ R[x] be non-zero polynomials in x. Then 1. deg(pq) = deg(p) + deg(q), 2. R[x]× ≡ R×, 3. R[x] is an integral domain. Since R ⊂ R[x] any zero divisors of R are also zero divisors of R[x] . If S is a subring of R, then S[x] is a subring of R[x] . Formal power series rings. (See Exercise 7.2.3) 1. R[[x]] is a commutative ring with 1R[[x]] . P∞ k × 2. (1 − x) k=0 1x = 1 shows that (1 − x) ∈ R[[x]] . P∞ k × × 3. k=0 a(k)x ∈ R[[x]] ⇔ a(0) ∈ R . Matrix rings. 1. If R is nontrivial and n ≥ 2 then Mn(R) is noncommutative and has zero divisors. Example: If ab 6= 0 a 0 0 b 0 ab 0 0 A = 6= 0,B = 6= 0, AB = 6= 0, BA = = 0 0 0 0 0 0 0 0 0 2. R is isomorphic to a subring of Mn(R) via R 3 a 7→ diag(a) ∈ Mn(R) . × 3. If R is commutative and has 1 ∈ R then Mn(R) = GL(R) the general linear group of degree n over R. 4. If S is a subring of R then Mn(S) is a subring of Mn(R) . Group rings. 1. RG is commutative if and only if G is commutative. a, if h = 1 ; 2. R is a subring of RG via R 3 a 7→ h 7→ G ∈ RG . 0R, if h 6= 1G. 1 , if h = g; 3. G is isomorphic to a subgroup of RG× via R 3 g 7→ h 7→ R ∈ RG× . 0R, if h 6= g. 4. If |G| > 1 then there exits g ∈ G with |g| = m > 1 . The computation (1 − g)(1 + g + g2 + ... + gm−1) = 1 − gm = 1 − 1 = 0 shows that 1 − g is a zero divisor of RG . 5. If S is a subring of R then SG is a subring of RG ..