Construction and Properties For Commutative Rings
Since the product of units is a unit and since ring homomorphisms respect products, we may and will assume that S is a submonoid of the multiplicative monoid of R, i.e. 1 is in S and for s and t in S we also have st in S. A subset of R with this property is called a multiplicatively closed set or, shortly, multiplicative set.
The assumption that S is multiplicatively closed is usual. If S is not multiplicatively closed, it suffices to replace it by its multiplicative closure consisting in the set of the products of elements of S. This does not change the result of the localization. The fact of talking of "a localization with respect of the powers of an element" instead of "a localization with respect to an element" is an example of this. Therefore we suppose S multiplicatively closed in what follows.
Read more about this topic: Localization Of A Ring
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