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.
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 . 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.
Definitions and Characterizations
We begin by giving a certain universal mapping characterization of product rings. Namely, let be a set of rings. We say that a ring satisfies the is a product of if there exists a set of epimorphisms such that for any ring and any set of morphisms there exists unique morphism such that . The important thing to note is:
Theorem: Let be a set of rings. Then, any two products of are isomorphic.
Proof: Let and be two products of with the associated sets of epimorphisms and . We first note by definition that since is a set of maps there exists a unique map such that and similarly there exists a unique map such that . We thus find that and similarly . Since both and are both epic (epimorphism) we may conclude that from the first of these equalities that and from the second that . Thus, we may conclude that and are isomorphisms with .
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 we define the the product of to be the set (with the usual function definition) with the addition of in to be and the product to be equal to . We define the canonical projections to be the map . Depending upon the connotation this map may also be called the evaluation homomorphism and is denoted . What we now note is that the product is a product. Indeed, the canonical projections are epimorphisms and evidently given a set of morphisms the function given by has the property that 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 the product with the set 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 ) 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 we denote by when convenient.
It’s important to note that if each ring is unital then the product is naturally unital with unit or in the finite case .
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 and and a set of morphisms we can form the product morphism
In other words, the “coordinate” is acted on by . To see precisely what this means it may be helpful to consider the finite case when given maps and we have that . It’s evident that the product of morphisms is itself a morphism. It’s also evident that
With this in mind we are in good position to prove:
Theorem: Let be a collection of rings and for each . Then, is an ideal of .
Proof: Let be the usual morphism and consider the morphism . By previous observation then we have that the kernel of this map is equal to which is equal to . Since we know the kernel of ring morphisms are ideals the conclusion follows.
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:
Theorem: Let be a set of unital rings. Then, all the ideals of are of the form for an ideal of if and only if .
Proof: The reason why all the ideals of say are of the form is as follows. We first note that if is an ideal then is an ideal since we know ideals are preserved under epimorphisms. We thus have that . To prove the reverse inclusion we note that if then there exists tuples (where just denotes some arbitrary element–i.e. we know there exists some tuple in which has in the coordinate and it’s irrelevant what it is). Now, since is an ideal we have that . Thus, and so the opposite inclusion holds. This half of the theorem then follows.
Conversely, suppose that is infinite and let be equal to all elements of 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.
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.