Definition 1

Let $A$ be a ring. A subset $S$ of $A$ is called multiplicative if every finite product of elements of $S$ belongs to $S$.

This is the same as saying that $1 \in S$ and that the product of two elements of $S$ belong to $S$.

Examples (1) For every $a \in A$, the set of $a^n$, where $n \in \mathbf{N}$, is a multiplicative subset of $A$. (2) Let $p$ be an ideal of $A$. For $A - p$ to be a multiplicative subset of $A$, it is necessary and sufficient that $p$ be prime. (3) The set of elements of $A$ which are not divisors of zero is a multiplicative subset of $A$.

(4) If S and T are multiplicative subsets of A, the set ST of products st, where s ∈ S and t ∈ T, is a multiplicative subset.

(5) Let G be a directed set (with respect to the relation ⊂ ) of multiplicative subsets of A. Then T = ⋃_{S ∈ G} S is a multiplicative subset of A, as any two elements of T belong to some subset S ∈ G, hence their product belongs to T.

(6) Every intersection of multiplicative subsets of A is a multiplicative subset.

For every subset S of a ring A, there exist multiplicative subsets of A containing S, for example A itself. The intersection of all these subsets is the smallest multiplicative subset of A containing S; it is said to be generated by S. It follows immediately that it is the set consisting of all the finite products of elements of S.