Abstract Nonsense

Crushing one theorem at a time

Localization (Pt. III)


Point of Post: This post is a continuation of this one.

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October 7, 2011 Posted by | Algebra, Ring Theory | , , , , , , , , , | 1 Comment

Localization (Pt. II)


Point of Post: This post is a continuation of this one.

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October 7, 2011 Posted by | Algebra, Ring Theory | , , , , , , , , , | 5 Comments

Polynomial Rings (Pt. III)


Point of Post: This post is a continuation of this one.

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July 19, 2011 Posted by | Algebra, Ring Theory | , , , , , , , | 3 Comments

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

Third Ring Isomorphism Theorem


Point of Post: In this post we prove the third ring isomorphism theorem which takes the form \left(R/\mathfrak{b}\right)/\left(\mathfrak{a}/\mathfrak{b}\right)\cong R/\mathfrak{a}.

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Motivation

Similar to the second ring isomorphism theorem the third isomorphism theorem has an almost immediate proof from the first ring isomorphism theorem (which, of course, both really should harken back to the proofs of the first and second group isomorphism theorems since the ring isomorphism theorems are generalizations (for abelian groups at least) if we define the trivial product on G given by ab=0 for all a,b\in G). Thus, just like the second ring isomorphism theorem the real meat of the theorem, the real thing that need be considered is precisely how the question “is (R/\mathfrak{a})/(\mathfrak{a}/\mathfrak{b})\cong R/\mathfrak{b}?’ naturally comes up in our study of rings. So, suppose for a second that we have some ring R and two R-ideals \mathfrak{a} and \mathfrak{b} with \mathfrak{a}\supseteq\mathfrak{b}. We first note that \mathfrak{a}/\mathfrak{b}=\left\{a+\mathfrak{b}:a\in\mathfrak{a}\right\}\subseteq \left\{r+\mathfrak{b}:r\in R\right\}=R/\mathfrak{b} and moreover that \mathfrak{a}/\mathfrak{b} is an ideal of R/\mathfrak{b} since it’s the image of \mathfrak{a} under the canonical projection \pi:R\twoheadrightarrow R/\mathfrak{b} and we know that ideals are preserved under epimorphisms. So, it makes sense to consider the quotient ring \left(R/\mathfrak{b}\right)/(\mathfrak{a}/\mathfrak{b}). That said, the elements of this quotient ring are, for lack of a better word, horrific. I mean, the general element looks like \left(r+\mathfrak{b}\right)+\mathfrak{a}/\mathfrak{b}. A wish would be that there was some nice way to describe this ‘double quotient ring’ as something nicer, something less unwieldy. The third ring isomorphism theorem grants us this wish by telling us that the \mathfrak{b}‘s “cancel” in the sense that (R/\mathfrak{b})/(\mathfrak{a}/\mathfrak{b})\cong R/\mathfrak{a}. To see the intuition of this theorem let’s look at the case when R=\mathbb{Z}. We have by previous comment that the ideals of \mathbb{Z} are all of the form n\mathbb{Z} for some n\in\mathbb{N}\cup\{0\}. So, suppose we had two \mathbb{Z}-ideals, say n\mathbb{Z} and m\mathbb{Z}. We note that evidently n\mathbb{Z}\supseteq m\mathbb{Z} if and only if n\mid m.We then note that what n\mathbb{Z}/m\mathbb{Z} ‘looks like’ (by definition)  the set \left\{\cdots,-n+\mathbb{Z}/m\mathbb{Z},0+\mathbb{Z}/m\mathbb{Z},n+\mathbb{Z}/m\mathbb{Z},\cdots\right\}. So, now let’s list the elements of \left(\mathbb{Z}/m\mathbb{Z}\right)/\left(n\mathbb{Z}/m\mathbb{Z}\right). They look like:

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July 1, 2011 Posted by | Algebra, Ring Theory | , , , , , , , , | 2 Comments

Ideals and Homomorphisms


Point of Post: In this post we explore some of the relations between ideals and homomorphisms, things like preimages of ideals, and images of epimorphisms are ideals, etc. We will not discuss the isomorphism theorems in this post, that will be covered later.

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Motivation

So far in our discussion of ideals we’ve completely ignored the interaction between ideals and homomorphisms (except that every kernel is an ideal). What kind of relationships should we expecdt to have? Since ideals are supposed to play the role for rings that normal subgroups played for groups it seems natural to look at the facts that held there. Things such as the preimage of an ideal is an ideal, etc. We also explore some of the ramifcations of these relationships for morphisms between certain kinds of rings, in particular for morphisms k\to k' for fields k,k'.

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June 24, 2011 Posted by | Algebra, Ring Theory | , , , , , , , | 3 Comments

Generated Ideals and the Lattice of Ideals (Pt. I)


Point of Post: In this post we discuss the notion of generated ideals from subsets of a ring as well as the fact that the ideals of a ring form a complete modular lattice.

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Motivation

Much as was the case for normal subgroups of a group it’s true that there is some lattice like structure for ideals. In this post we shall describe the notion of  generated ideals and use it to prove that the poset \left(\subseteq,\mathfrak{L}\left(R\right)\right) of ideals of R is a complete modular lattice .

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June 22, 2011 Posted by | Algebra, Ring Theory | , , , , , , , , , | 5 Comments

Definition and Basics of Ideals


Point of Post: In this post we define left, right, and two-sided ideals.

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Motivation

As was alluded to when we discussed subrings, subrings themselves aren’t the important thing when discussing rings, in the same way that it’s not subgroups but normal subgroups that play a pivotal role in group theory. In particular, what we’d eventually like to do is to define the quotient objects for rings–the ‘quotient rings’. But, as normal we can’t quotient out by any old subset, or even any old subring, of a ring. It turns out, much the same as the case for groups that what defines these special subrings is that they are the kernels of ring homomorphisms. But, this is all in the future. What we shall show in this post is that kernels of ring homomorphisms satisfy a certain ‘absorption’ property and then define these analogs of normal subgroups as being subrings that satisfy this property

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June 21, 2011 Posted by | Algebra, Ring Theory | , , , , , , , | 13 Comments