Algebraic definitions

The pair <S, +> is said to be a group if:

• for every a, b, and c in S, (a + b) + c = a + (b + c);
• there is an identity element 0 in S such that for every a in S, 0 + a = a + 0 = a; and
• for every a in S, there is an inverse -a such that a + -a = -a + a = 0.

If we use the symbol * as the operator in the group, we usually call the identity 1 and write the inverse of a as a^-1.

A group <S, +> is said to be commutative if for every a and b in S, a + b = b + a. We also call such a group an abelian group.

Suppose S has n elements. Then we say the group <S, +> is of order n.

Let <S, +> be a group of order n. We abbreviate the sum of i a's, a + a + ... + a, as ia. (Likewise, we abbreviate the product of i a's, a * a * ... * a, as aⁱ.) <S, +> is said to be cyclic if there is a generator k in S such that nk = 0 and for every positive integer i < n, ik ≠ 0. In this case, it is easily seen that for each a in S, there is exactly one positive integer i ≤ n such that ik = a.

Let <S, +> and <S', +> be groups such that S' is a subset of S. Then <S', +> is said to be a subgroup of <S, +>.

The triple <S, +, *> is said to be a ring if:

• <S, +> is an abelian group;
• for every a, b, and c in S, (a * b) * c = a * (b * c); and
• for every a, b, and c in S, a * (b + c) = a * b + a * c and (b + c) * a = b * a + c * a.

If there is an element 1 in S such that for every a in S, 1 * a = a * 1 = a, then the ring <S, +, *> is said to be a ring with unit element.

If for every a and b in S, a * b = b * a, then the ring <S, +, *> is said to be a commutative ring.

Let 0 be the additive identity (i.e., the identity of <S, +>) of the ring <S, +, *>. If <S-{0}, *> is an abelian group, then <S, +, *> is said to be a field.