Suggestions for further topics in Commutative Algebra
This answer may be viewed as a bit odd and out of fashion by some, but one direction in which you could go--which intersects with commutative algebra, algebraic number theory, and discrete mathematics--is factorization theory.
Back when you were memorizing your multiplication tables in elementary school, the Fundamental Theorem of Arithmetic made that relatively easy in that there was only one way to break down a positive integer (greater than 1) into primes (which can be thought of as "building blocks", in a sense).
The generalization of this property that is discussed in the standard undergraduate abstract algebra sequence is, of course, the concept of the unique factorization domain (UFD), also called a factorial domain. The canonical example of an integral domain that does not enjoy unique factorization is the following example in $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5}\,\vert\,a,b\in \mathbb{Z}\}$:
$6=2\cdot3 = (1-\sqrt{-5})(1+\sqrt{-5})$.
One can use the standard norm function for rings of algebraic integers to show that 2,3, and $1\pm \sqrt{-5}$ are all irreducible in $\mathbb{Z}[\sqrt{-5}]$, and that the two factorizations of 6 above are distinct.
What's interesting, though, is that it turns out that while nonunit elements of $\mathbb{Z}[\sqrt{-5}]$ may not factor uniquely into irreducibles, any two factorizations of a fixed nonzero nonunit element have the same number of irreducible factors. Such a domain is called a half factorial domain (or HFD). In a short, two page paper in the Proceedings of the AMS, L. Carlitz characterized all HFDs amongst rings of algebraic integers via the (ideal) class group. Such rings have unique factorization precisely when the class group is trivial and are HFDs precisely when the class group is isomorphic to $\mathbb{Z}_2$. So, in the context of rings of algebraic integers, the size of the class group gives a measurement of "how far" we are from unique factorization.
Since Carlitz's seminal 1960 paper, there has been a lot of work done in factorization theory (and yes, this was my area of research while I was in academia, so I'm not exactly unbiased). Currently, the only book on the topic is Geroldinger and Halter-Koch's Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Make no mistake, this book is an extremely dense read, but it's very thorough and has an excellent bibliography.
Other sources (some of which are a bit more accessible to the newcomer) include:
- The stuff on Scott Chapman's website. Scott has run a wildly successful REU program, and many of the projects have been in factorization theory.
- Some research papers and course materials on Jim Coykendall's website.
Well, here are some of my thoughts. Valuations is an important topic in its own right however you will find most elementary applications of this theory in algebraic number theory and perhaps theory of curves. So, the material contained in Atiyah Macdonald (and perhaps Matsumura) is standard (not very difficult). The more general setting of valuations is discussed in Bourbaki Commutative algebra however the presentation may seem too pedantic.
Homological Algebra on the other hand is a very very useful tool. It takes sometime to get an intuition and get a feel for the basic techniques (which may seem ad hoc). It is a central tool for algebraic geometry and commutative algebra itself.
Moreover things like flatness, regularity etc can be formulated in terms of derived functors of homological algebra. As an added bonus the same formalism (of homological algebra) applies to many other areas e.g group theory (group homology-cohomology),lie algebras, essentially any abelian category.
Some references of homological algebra (in some generality) will be:
Weibel's Book (it covers a lot of ground so might seem little un-motivated at places).
Cartan Eilenberg (wonderful book with a little outdated terminology)
Tohuku by Grothendieck.
Parts of Homological Algebra by Gelfand Manin.
Of course it is usually a good practice to come back to commutative algebra (after a little excursion into homological world) and re-examine the basic homological invariants again.