Determine whether the set S a b Z times Z a b 0 is a sub

Determine whether the set S = {(a, b) Z times Z | a + b = 0} is a subring of Z times Z. If it is, prove so, and if it is not, give an example that demonstrates so.

Solution

A set R is called a ring, it satisfies the following conditions:

·         (p + q) + r = p + (q + r) for all p, q, r R (+ is associative).

· p + q = q + p for all p, q R (+ is commutative).

·         There is an element 0 in R such that p + 0 = p for all p R (0 is the additive identity).

·         For each p R there exists p in R such that p + (p) = 0 (p is the additive inverse of p).

·         (p . q)· r = p · (q · r) for all p, q, r R (· is associative).

·         There is an element 1 in R such that p · 1 = p = 1 · p = p for all p R (1 is the multiplicative identity).

·         p (q + r) = (p · q) + (p · r) for all p, q, r R (left distributivity).

·         (q + r) · p = (q · p) + (r · p) for all p, q, r R (right distributivity).·

Also, a non empty subset S of R is called a sub ring of R if (S, +, . ) is a ring.

Here S = {(a, b) Z x Z : a + b = 0}. While checking whether S satisfies all the above conditions , we find that (1, 1) is the multiplicative identity in Zx Z, but it does not belong to S as 1 + 1 0. Hence S is not a sub ring of Z x Z.