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Let $S$ be a subset of the ring $R$; we say that $S$ is multiplicative if

  (a) $0 \notin S$,
  (b) $1 \in S$, and
  (c) whenever $a,b\in S$, we have $ab \in S$.

We can merge (b) and (c) by stating that $S$ is closed under multiplication, if we regard $1$ as the empty product.

Here are some standard examples of multiplicative sets.

  1. The set of all nonzero elements of an integral domain.
  2. The set of all nonzero elements of a commutative ring $R$ that are not zero divisors.
  3. $R \setminus P$, where $P$ is a prime ideal of the commutative ring $R$.

I think 3 is not a correct example as $R \setminus P$ is an integral domain and has zero element $P$. It should be $R \setminus P$ except $P$. Is it right?

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A good exercise to see the link between prime ideals and multiplicative sets: Prove that an ideal $I$ of a ring $R$ is a prime ideal if and only if $R\setminus I$ is a multiplicative set. – Ragib Zaman Jun 28 '13 at 9:28
up vote 6 down vote accepted

The notation $\,R\backslash P\,$ means exactly "all the elements in the set $\;R\;$ except those belonging to $\;P\;$" .

I think you confused this with the quotient ring $\;R/P\;$ ...

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I see. Thank you – Jebei Jun 28 '13 at 8:32
@DonAntonio: This is emphasized by the fact that the correct LaTeX symbol is \setminus :-) – Zev Chonoles Jun 28 '13 at 8:36
Indeed @ZevChonoles , yet the resulting symbols are extremely alike: $$R\backslash P\;,\;\;R\setminus P$$ It's almost the same with \backslash and \setminus – DonAntonio Jun 28 '13 at 8:38

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