multiplicatively closed
Let $R$ be a ring. A subset $S$ of $R$ is said to be multiplicatively closed if $S\ne \mathrm{\varnothing}$, and whenever $a,b\in S$, then $ab\in S$. In other words, $S$ is a multiplicative set where the multiplication^{} defined on $S$ is the multiplication inherited from $R$.
For example, let $a\in R$, the set $S:=\{{a}^{i},{a}^{i+1},\mathrm{\cdots},{a}^{n},\mathrm{\cdots}\}$ is multiplicatively closed for any positive integer $i$. Another simple example is the set $\{1\}$, if $R$ is unital.
Remarks. Let $R$ be a commutative ring.

•
If $P$ is a prime ideal^{} in $R$, then $RP$ is multiplicatively closed.
 •

•
In particular, assuming $1\in R$, any ideal maximal with respect to being disjoint from $\{1\}$ is a maximal ideal^{}.
A multiplicatively closed set $S$ in a ring $R$ is said to be saturated^{} if for any $a\in S$, every divisor^{} of $a$ is also in $S$.
In the example above, if $i=1$ and $a$ has no divisors, then $S$ is saturated.
Remarks.

•
In a unital ring, a saturated multiplicatively closed set always contains $U(R)$, the group of units of $R$ (since it contains $1$, and therefore, all divisors of $1$). In particular, $U(R)$ itself is saturated multiplicatively closed.

•
Assume $R$ is commutative^{}. $S\subseteq R$ is saturated multiplicatively closed and $0\notin S$ iff $RS$ is a union of prime ideals in $R$.
Proof.
This can be shown as follows: if let $T$ be a union of prime ideals in $R$ and $a,b\in RT$. if $ab\notin RT$, then $ab\in P\subseteq T$ for some prime ideal $P$. Therefore, either $a$ or $b\in P\subseteq T$. This contradicts the assumption^{} that $a,b\notin T$. So $RT$ is multiplicatively closed. If $ab\in RT$ with $a\notin RT$, then $a\in P\subseteq T$ for some prime ideal $P$, which implies $ab\in P\subseteq T$ also. This contradicts the assumption that $ab\notin T$. This shows that $RT$ is saturated. Of course, $0\notin RT$, since $0$ lies in any ideal of $R$.
Conversely, assume $S$ is saturated multiplicatively closed and $0\notin S$. For any $r\notin S$, we want to find a prime ideal $P$ containing $r$ such that $P\cap S=\mathrm{\varnothing}$. Once we show this, then take the union $T$ of these prime ideals and that $S=RT$ is immediate. Let $\u27e8r\u27e9$ be the principal ideal^{} generated by $r$. Since $S$ is saturated, $\u27e8r\u27e9\cap S=\mathrm{\varnothing}$. Let $M$ be the set of all ideals containing $\u27e8r\u27e9$ and disjoint from $S$. $M$ is nonempty by construction, and we can order $M$ by inclusion. So $M$ is a poset and Zorn’s lemma applies. Take any chain $C$ in $M$ containing $\u27e8r\u27e9$ and let $P$ be the maximal element^{} in $C$. Then any ideal larger than $P$ must not be disjoint from $S$, so $P$ is prime by the second remark in the first set of remarks. ∎

•
The notion of multiplicative closure^{} can be generalized to be defined over any nonempty set with a binary operation (multiplication) defined on it.
References
 1 I. Kaplansky, Commutative Rings. University of Chicago Press, 1974.
Title  multiplicatively closed 

Canonical name  MultiplicativelyClosed 
Date of creation  20130322 17:29:15 
Last modified on  20130322 17:29:15 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 13B30 
Classification  msc 16U20 
Synonym  saturated 
Related topic  MSystem 
Defines  saturated multiplicatively closed 