Wednesday, July 01, 2009
Rings
Every time I read something that pertains to any kind of ring, there's always a necessary section at the beginning stipulating that such-and-such a ring has an identity, that the mappings under consideration must preserve that identity, and so forth. Can someone fill me in as to whether there's some area of study in which people use rings without 1 so extensively that it makes up for the extreme annoyance of dealing with such requirements everywhere else? As far as I can tell, rings without 1 occur basically as often as these things called "monoids" or "groupoids" -- more than never, but certainly not often enough to merit such a basic name, or take up space in textbooks and brains. Can someone set me straight here?
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5 comments:
Rings without an identity show up more frequently than you might expect. An easy set of examples come from various rings of functions on noncompact spaces subject to vanishing conditions at infinity (for instance, the ring of continuous compactly supported functions on the real line). A more important example arises in Fourier analysis. The space L^1(R) is not closed under multiplication. It is, however, closed under convolution, and it is an easy exercise to show that the resulting ring has no identity.
So am I correct in receiving the impression that this is something mainly seen in functional analysis?
Another place you might come across rings without units is as follows. Let R be a ring with a unit. Define S to be the direct SUM of infinitely many copies of R. An element of S is thus a sequence (r_1,r_2,...) of elements of R, only finitely many of which are nonzero. Exercise : S has no unit.
However, most of the natural examples that come to my mind immediately come from analysis. This might just reflect my biases as a mathematician :). The four main "users" of ring theory that I'm pretty acquainted with are (1) Algebraic geometric (2) Algebraic number theory (3) Representation theory (4) Functional analysis.
Of these four areas, only (4) makes serious use of rings without units.
In "pure algebra", not having a unit is pretty harmless, as there is a natural way to embed any ring without unit into a ring with unit (this is an easy exercise -- just adjoint a element to serve as a unit, and check the axioms).
Like an idiot, I forgot the easiest examples! Let n>1 be an integer. Then the subring nZ of the integers Z has no unit.
More generally, one could replace Z by any ring that is not a field and nZ by any nontrivial ideal.
Of course, these rings are really best viewed as modules over the original ring.
My final comment (really!). I forgot to mention that adding a unit to a ring requires that the ring have no zero-divisors.
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