Definitions for CM, etc

CM ring - CM
CM-mod over local ring - Regular Seq.
Regular local ring - Reg. Seq.

(A,M) flat then (B,M*B) flat

new CM

Chase(Over a Noetherian ring, direct product of flat modules is flat.)

CM

M->S^-1M being injective or surjective iff s is M-regular

Post-c.i.

Extension & Contraction of scalars for modules again

Including localization

post-c.i.

Localization, universal property for modules

DVR & Dedekind

Also see Jordan-Holder

Irreducible ideal being primary (commutative ring)

Ass(Redux), dim(M)

0->M->M->M->0 ACC,DCC, f.g. module over Noeth is Noeth

NC-IV

ann(m), noncommutative

NC-III

Injective module redux

Restriction & extension of scalars

Projectivity of module M and Ext^1(M,N)=0 for all f.g. N

Ext,Tor, pd, id

Also Eisenbud's projective characterization from appendix

simple module

Injective modules

Jordan-Holder

Isomorphic to R/m (NC-III, Jacobson)

Socle review

Injective Modules

Maximal subset of direct sum

Injective Modules

Matrix ring example

QLOCAL

Submodule definition

NON-COMM-PROP

(r)_L, (r)_R, (r)

NON-COMM-PREL

If I is a left ideal, so is IS for any subset S

NON-COMM-PROP

aM, Im, IM, ann(M), etc. (commutative)

Commutative - Injective modules, under APPENDIX

Non-commutative - NON-COMM-II

Localization again (S,T, transitivity, p followed by q)

Regular rings (Also localization of A[x])

Jordan-Holder

Saturated mult subset

Module Category & Rank & Matsumura

Module category revisited (II) (Localization, extension, etc.)

Jordan-Holder

First module category -- Matsu&

Filtration of modules, support, Ass

Ext, Tor, pd, id

Also Ass(Redux)