## Maximal Ideals (Pt. I)

**Point of Post: **In this post we discuss the notion of maximal ideals, give several characterizations for commutative unital rings, etc.

*Motivation*

Just as is the case for prime ideals there are two natural ways to motivate the introduction to maximal ideals. That said, unlike the case for prime ideals where any kid off the street can relate to their importance by pointing the value of primes in the value of maximal ideals is a slightly more delicate issue, but one which plays a pivotal role in much of advanced ring theory. In fact, a very common technique in more advanced ring theory goes as follows:

where a *local ring* is a ring with a unique maximal ideal , often denoted . The reason for this (analogous to the embedding of integral domains for fields) is that, as we shall (maybe not so soon) see, local rings are particularly simple.

So what are these ‘maximal’ ideals that are so important? As stated there are two natural ways to motivate why anyone would ever even consider such a kind of ideal. As in the case with prime ideals we start with a simple minded, but important motivating factor. In fact, this motivation is in direct parallel with the first motivating factor for prime ideals. Namely, for a commutative unital ring we decided that prime ideals were the answer to “what kind of ideal is characterized by having be an integral domain. Well, while I can’t speak for the whole of the mathematical community, I feel that it’s not natural to ask when is an integral domain. No, I believe that most people would want the whole shebang and ask when is a field. Well, the answer to this question is “when is a maximal ideal”. Well, what precisely are ‘maximal ideals’? Well, let’s see if we can take them, by definition, to be ideals whose attendant quotient ring is a field and see if we can work backwards towards their ‘real’ definition. So, we begin by picking a characterization of fields that is particularly amenable to our current situations. Considering we are dealing strictly with ideals it makes sense that this would be the characterization of fields in terms of ideals. In particular, one can recall that if is commutative and unital then is a field if and only if all it’s ideals are trivial or non-propert. In simpler terms, will be a field precisely when the only ideals of are and . Thus, an ideal of will be ‘maximal’ if and only if ideal lattice of looks like

But, now we’re cooking with fire since the fourth isomorphism theorem tells us this is the case only when the ideal structure of looks like

Or, in other words, will have the desired ideal lattice if and only if is maximal (in terms of containment) amongst the proper ideals of . *This* is the definition of maximal that we shall take. Namely, for us an ideal (appropriately dubbed) will be called maximal when implies and .

The second motivation is one which, while equally important, doesn’t quite have the same verve as the previous motivation. Namely, a heuristic statement with some real truth behind it, is that a ring with a particularly simple lattice structure has a particularly simple overall ring structure. Thus, to find examples of rings which are easy to deal with, a sensible thing to do is to single out what is simple in lattice land and then transfer this back over to ring land. In particular, it’s not hard to see (if one draws a few examples of Hasse diagrams) that rings with few maximal ideals are ‘simple’.

*Maximal Ideals*

Let be a ring. We define a proper ideal of to *maximal *if whenever is an ideal of such that then . In other words, is a maximal ideal if it’s maximal in the poset of all proper ideals of . We denote the set of all maximal ideals of a ring by . As noted above in the introduction, we have by the fourth isomorphism theorem that:

**Theorem: ***Let be a commutative unital ring. Then, if and only if is a field.*

To see how this could fail for non-unital rings one can easily see that is maximal in yet is not a field. That said, it obviously fails for a stupid reason, the idea is still there. In particular, if one defines a *simple ring *to be a ring with only trivial ideals ( and ) then it’s evident that a more general version of the above says:

**Theorem: ***An ideal in an ideal is maximal if and only if is simple.*

**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.

3. Reid, Miles. *Undergraduate Commutative Algebra*. Cambridge: Cambridge UP, 1995. Print.

[…] be a field and let be irreducible. We know then that is a maximal ideal and thus is a field, which can be thought of as an extension of via the map sending . In fact, […]

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