"I need to brush up on some Pure Maths for a thing, and I find I'm pretty rusty. Can any of m'Maths friends recommend a good book on rings and ideals, and that end of algebra?"
@ColinTheMathmo Thanks all, this is great stuff. I've ordered a copy of The Cheapest One, and will head to the library tomorrow to check out the others.
@ColinTheMathmo I wrote mine, (mostly) commutative algebra, with part of this profile in mind. My Field guide on algebra goes towards Galois theory but has less general algebra in it, it tries to bypass that.
@ColinTheMathmo What I have on my shelf is Hartley & Hawkes, "Rings, Modules and Linear Algebra". Fairly short. Looks pretty clear from a quick glance. Seems to be out of print but available second-hand at reasonable prices.
Part I (~90pp): rings and modules: definitions and examples of rings; subrings, homomorphisms, ideals; direct sums, polynomial rings, etc.; integral domains and factorization; modules (inc. homomorphisms, quotients, direct sums); a few special classes of modules.
Part II (~45pp): decomposition theorems for f.g. modules over a p.i.d.
Part III (~50pp): applications to groups and matrices: f.g. abelian groups as ℤ-modules, linear transformations, matrices, canonical forms; computation of canonical forms.
I somehow don't have a copy of Herstein to compare with.
@ColinTheMathmo Herstein's Topics in Algebra. It does have a chapter on group theory before the Rings and Ideals stuff, but it's still my top recommendation.
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