Homework nr.2 examples of rings

(by Robert_Tomaszewicz)

Definition 1.1. A ring R is a set with two binary operations, + and · , satisfying: ring

(1)   (R,+) is an abelian group,

(2)    R is closed under multiplication, and (ab)c = a(bc) for all a, b, c ∈ R,

(3)    a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c ∈ R.

Example 1.2 (Examples of rings). 1. Z, Q, R, C.

2. 2Z – even numbers. Note that 1 ∈ 2Z.

3. Matn(R) = {n × n-matrices with real entries}
   In general AB < > BA.

   A ring R is called commutative if ab = ba for all a, b ∈ R.

4. Fix m, a positive integer. Consider the remainders modulo m: 0, 1, ...,m − 1.
 
Notation. Write n for the set of all integers which have the same remainder as n
n when divided by m. This is the same as {n + mk | k ∈ Z}. Also, n1 + n2 =
n1 + n2, and n1 · n2 = n1n2. The classes 0, 1, . . . ,m − 1 are called residues
modulo m.
The set  {0, 1, ...,m − 1}  is denoted by Zm or by Z/m or by Z/mZ.

5. The set of polynomials in x with coefficients in Q (or in R or C)

                                  {a0 + a1x + ... + anx2 | ai ∈ Q ,= Q[x]}
                with usual addition and multiplication. If an < > 0 then n is the degree of the
                polynomial.

Definition 1.3. A subring of a ring R is a subset which is a ring under the same subring
addition and multiplication.

Proposition 1.4. Let S be a non-empty subset of a ring R. Then S is a subring of
R if and only if, for any a, b ∈ S we have a + b ∈ S, ab ∈ S and −a ∈ S.

Proof. A subring has these properties. Conversely, if S is closed under addition and
taking the relevant inverse, then (S,+) is a subgroup of (R,+) (from group theory).
S is closed under multiplication.
Associativity and distributivity hold for S because they hold for R.