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[parent] multiplicatively closed (Definition)

Let $ R$ be a ring. A subset $ S$ of $ R$ is said to be multiplicatively closed if $ S\ne \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:=\lbrace a^i, a^{i+1}, \cdots, a^n, \cdots \rbrace$ is multiplicatively closed for any positive integer $ i$. Another simple example is the set $ \lbrace 1\rbrace$, if $ R$ is unital.

Remarks. Let $ R$ be a commutative ring.

  • If $ P$ is a prime ideal in $ R$, then $ R-P$ is multiplicatively closed.
  • Furthermore, an ideal maximal with respect to the being disjoint from a multiplicative set not containing 0 is a prime ideal.
  • In particular, assuming $ 1\in R$, any ideal maximal with respect to being disjoint from $ \lbrace 1\rbrace$ 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 $ R-S$ 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 R-T$. if $ ab\notin R-T$, 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 $ R-T$ is multiplicatively closed. If $ ab \in R-T$ with $ a\notin R-T$, 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 $ R-T$ is saturated. Of course, $ 0\notin R-T$, 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=\varnothing$. Once we show this, then take the union $ T$ of these prime ideals and that $ S=R-T$ is immediate. Let $ \langle r\rangle$ be the principal ideal generated by $ r$. Since $ S$ is saturated, $ \langle r\rangle\cap S=\varnothing$. Let $ M$ be the set of all ideals containing $ \langle r\rangle$ and disjoint from $ S$. $ M$ is non-empty 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 $ \langle r\rangle$ 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. $ \qedsymbol$

  • The notion of multiplicative closure can be generalized to be defined over any non-empty set with a binary operation (multiplication) defined on it.

Bibliography

1
I. Kaplansky, Commutative Rings. University of Chicago Press, 1974.



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See Also: $m$-system

Other names:  saturated
Also defines:  saturated multiplicatively closed

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Cross-references: binary operation, closure, multiplicative, prime, maximal element, chain, Zorn's lemma, poset, inclusion, order, generated by, principal ideal, implies, union, iff, commutative, group of units, contains, unital ring, divisor, maximal ideal, disjoint, ideal, prime ideal, commutative ring, unital, simple, integer, positive, multiplication, multiplicative set, subset, ring
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This is version 3 of multiplicatively closed, born on 2007-08-18, modified 2007-10-13.
Object id is 9875, canonical name is MultiplicativelyClosed.
Accessed 1020 times total.

Classification:
AMS MSC: 13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)
 16U20 (Associative rings and algebras :: Conditions on elements :: Ore rings, multiplicative sets, Ore localization)
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