Multiplicatively closed set

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a multiplicatively closed set (or multiplicative set) is a <a href="/facts/Subset/SJGERmbA">subset</a> S of a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> R such that the following two conditions hold:

<ul><li>
 
 
 
 1
 ∈
 S
 
 
 {\displaystyle 1\in S}
 
,</li>
<li>
 
 
 
 x
 y
 ∈
 S
 
 
 {\displaystyle xy\in S}
 
 for all 
 
 
 
 x
 ,
 y
 ∈
 S
 
 
 {\displaystyle x,y\in S}
 
.</li></ul>
In other words, S is <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closed</a> under taking finite products, including the <a href="/facts/Empty_product/mj7nhGeD">empty product</a> 1.
Equivalently, a multiplicative set is a <a href="/facts/Submonoid/6AHve7p8">submonoid</a> of the multiplicative <a href="/facts/Monoid/6AHve7p8">monoid</a> of a ring.
Multiplicative sets are important especially in <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a>, where they are used to build <a href="/facts/Localization_of_a_ring/rjevzBJf">localizations</a> of commutative rings.
A subset S of a ring R is called saturated if it is closed under taking <a href="/facts/Divisibility_(ring_theory)/KQdsMrLw">divisors</a>: i.e., whenever a product xy is in S, the elements x and y are in S too.

Multiplicatively closed set open-in-new

Multiplicatively closed set