Abstract Nonsense

Crushing one theorem at a time

Product of Rings (Pt. I)


Point of Post: In this post we discuss the product of rings including the classification for ideals of finite products and the universal property of product rings.

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Motivation

As for groups, topological spaces, representations, in any category it is always useful to define product structures. So, in this post we shall describe what it means to form the ‘product ring’ of the set of rings \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}}. We will then show that similarly to quotient rings there is a universal mapping property which characterizes product rings. We will also discuss the situation of finding the ideals of a product of unital rings in terms of the ideals of the individual rings, and in fact characterize the difference infiinitely and finitely many factors in the product (i.e. whether we are taking the product of finitely many or infinitely many rings) based on the ideals of the product ring.

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Definitions and Characterizations

We begin by giving a certain universal mapping characterization of product rings. Namely, let \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} be a set of rings. We say that a ring R satisfies the is a product of \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} if there exists a set of epimorphisms \left\{p_\alpha:R\to R_\alpha\right\}_{\alpha\in\mathcal{A}} such that for any ring S and any set of morphisms \left\{f_\alpha:S\to R_\alpha\right\}_{\alpha\in\mathcal{A}} there exists unique morphism g:S\to R such that p_\alpha\circ g=f_\alpha. The important thing to note is:

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Theorem: Let \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} be a set of rings. Then, any two products of \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} are isomorphic.

Proof: Let P and P' be two products of \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} with the associated sets of epimorphisms \left\{p_\alpha\right\}_{\alpha\in\mathcal{A}} and \left\{p'_\alpha\right\}_{\alpha\in\mathcal{A}}. We first note by definition that since p'_\alpha:P'\to R_\alpha is a set of maps there exists a unique map f:P'\to P such that p_\alpha\circ f=p'_\alpha and similarly there exists a unique map g:P\to P' such that p'_\alpha\circ g=p_\alpha. We thus find that p_\alpha=p'_\alpha\circ g=p_\alpha\circ f\circ g and similarly p'_\alpha=p'_\alpha\circ g\circ f. Since both p_\alpha and p'_\alpha are both epic (epimorphism) we may conclude that from the first of these equalities that f\circ g=\text{id}_P and from the second that g\circ f=\text{id}_{P'}. Thus, we may conclude that f and g are isomorphisms with g=f^{-1}. \blacksquare

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We thus see that if (and in general this is a big if) products exist then they are, up to isomorphism, unique. Thus, it suffices to show that there actually exist models of products of a set of rings. In particular, given a set of rings \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} we define the the product of \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} to be the set \displaystyle \prod_{\alpha\in\mathcal{A}}R_\alpha (with the usual function definition) with the addition f+g of f,g in \displaystyle \prod_{\alpha\in\mathcal{A}}R_\alpha to be (f+g)(\alpha)=f(\alpha)+g(\alpha) and the product fg to be equal to (fg)(\alpha)=f(\alpha)g(\alpha). We define the canonical projections \displaystyle \pi_\beta:\prod_{\alpha\in\mathcal{A}}R_\alpha\to R_\beta to be the map \pi_\beta(f)=f(\beta). Depending upon the connotation this map may also be called the evaluation homomorphism and is denoted \text{ev}_\beta. What we now note is that the product is a product. Indeed, the canonical projections are epimorphisms and evidently given a set of morphisms f_\alpha:R\to R_\alpha the function \displaystyle f:R\to\prod_{\alpha\in\mathcal{A}}R_\alpha given by (f(r))(\beta)=f_\beta(r) has the property that \pi_\beta\circ f=f_\beta and evidently is the unique function to do this. Thus, when we think about products we shall be thinking about this model of a product for the infinite case. For the finite case there is a nicer way to think about the product of rings instead of the function definition. Namely, we identify for a finite set of rings \left\{R_1,\cdots,R_n\right\} the product \displaystyle \prod_{\alpha\in[n]}R_\alpha with the set R_1\times \cdots\times R_n=\left\{(r_1,\cdots,r_n):r_k\in R_k\right\} with coordinate wise multiplication and addition. It’s important to note that we arbitrarily chose the order the tuples appeared in (i.e. we chose the first tuple to contain elements of R_1) but since evidently any reordering of the factors is canonically isomorphic (prove the obvious identification is an isomorphism if you’ve never seen it done before!). In particular, if R_1=\cdots=R_n=R we denote R_1\times\cdots\times R_n by R^n when convenient.

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It’s important to note that if each ring is unital then the product is naturally unital with unit 1(\alpha)=1_\alpha or in the finite case 1=(1_1,\cdots,1_n).

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Product of Morphisms and the Ideals of a Product of Unital Rings

An obvious question now is, what are the ideals of a product? To answer this we first define an object which should be familiar to most people. Namely, suppose that have to sets of rings \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} and \left\{S_\alpha\right\}_{\alpha\in\mathcal{A}} and a set of morphisms \left\{f_\alpha:R_\alpha\to S_\alpha\right\} we can form the product morphism

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\displaystyle \prod_{\alpha\in\mathcal{A}}f_\alpha:\prod_{\alpha\in\mathcal{A}}R_\alpha\to\prod_{\alpha\in\mathcal{A}}S_\alpha

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given by

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\displaystyle \left(\left(\prod_{\alpha\in\mathcal{A}}f_\alpha\right)(g)\right)(\beta)=f_\beta(g(\beta))

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In other words, the \beta^{\text{th}} “coordinate” is acted on by f_\beta. To see precisely what this means it may be helpful to consider the finite case when given maps f:R_1\to S_1 and g:R_2\to S_2 we have that (f\times g)(r_1,r_2)=(f(r_1),f(r_2)). It’s evident that the product of morphisms is itself a morphism. It’s also evident that

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\displaystyle \ker\left(\prod_{\alpha\in\mathcal{A}}f_\alpha\right)=\prod_{\alpha\in\mathcal{A}}\ker f_\alpha

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With this in mind we are in good position to prove:

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Theorem: Let \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} be a collection of rings and \mathfrak{a}_\alpha\in\mathfrak{L}(R_\alpha) for each \alpha\in\mathcal{A}. Then, \displaystyle \prod_{\alpha\in\mathcal{A}}\mathfrak{a}_\alpha is an ideal of \displaystyle \prod_{\alpha\in\mathcal{A}}R_\alpha.

Proof: Let \pi_\alpha:R\to R/\mathfrak{a}_\alpha be the usual morphism and consider the morphism \displaystyle \prod_{\alpha\in\mathcal{A}}\pi_\alpha. By previous observation then we have that the kernel of this map is equal to \displaystyle \prod_{\alpha\in\mathcal{A}}\ker\pi_\alpha which is equal to \displaystyle \prod_{\alpha\in\mathcal{A}}\mathfrak{a}_\alpha. Since we know the kernel of ring morphisms are ideals the conclusion follows. \blacksquare

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What we now show is that the converse is a characterization of whether we are taking the product of infinitely many or finitely many unital rings. Namely:

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Theorem: Let \left\{R_\alpha\right\}_{\alpha\in\mathcal{A}} be a set of unital rings. Then, all the ideals of \displaystyle \prod_{\alpha\in\mathcal{A}}R_\alpha are of the form \displaystyle \prod_{\alpha\in\mathcal{A}}\mathfrak{a}_\alpha for \mathfrak{a}_\alpha an ideal of R_\alpha if and only if \#(\mathcal{A})<\infty.

Proof: The reason why all the ideals of say R_1\times\cdots\times R_n are of the form \mathfrak{a}_1\times\cdots\times\mathfrak{a}_n is as follows. We first note that if \mathfrak{a} is an ideal then \pi_j(\mathfrak{a})\overset{\text{def.}}{=}\mathfrak{a}_j is an ideal since we know ideals are preserved under epimorphisms. We thus have that \mathfrak{a}\subseteq\mathfrak{a}_1\times\cdots\times\mathfrak{a}_n. To prove the reverse inclusion we note that if r_j\in\mathfrak{a}_j then there exists tuples (r_1,\cdots,\ast),(\ast,r_2,\cdots,\ast),\cdots,(\ast,\cdots,r_n)\in\mathfrak{a} (where \ast just denotes some arbitrary element–i.e. we know there exists some tuple in \mathfrak{a} which has r_j in the j^{\text{th}} coordinate and it’s irrelevant what it is). Now, since \mathfrak{a} is an ideal we have that (0,\cdots,r_j,\cdots,0)=(\ast,\cdots,r_j,\cdots,\ast)(0,\cdots,1_j,\cdots,0)\in\mathfrak{a}. Thus, (r_1,\cdots,r_n)=(r_1,\cdots,0)+\cdots+(0,\cdots,r_n)\in\mathfrak{a} and so the opposite inclusion holds. This half of the theorem then follows.

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Conversely, suppose that \mathfrak{A} is infinite and let \mathfrak{a} be equal to all elements of \displaystyle \prod_{\alpha\in\mathcal{A}}R_\alpha with finite support (only finitely many non-zero coordinates). This is evidently an ideal but evidently not the product of ideals since it clearly isn’t even the product of sets. \blacksquare

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References:

1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

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July 11, 2011 - Posted by | Algebra, Ring Theory | , , , , , ,

11 Comments »

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