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
Famous quotes containing the words construction, properties and/or rings:
“Striving toward a goal puts a more pleasing construction on our advance toward death.”
—Mason Cooley (b. 1927)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“Ye say they all have passed away,
That noble race and brave;
That their light canoes have vanished
From off the crested wave;
That, mid the forests where they roamed,
There rings no hunters’ shout;
But their name is on your waters,
Ye may not wash it out.”
—Lydia Huntley Sigourney (1791–1865)