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Jul
27
comment Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.
Just as a point of order, I don't think algebraists in general are that rigid about notation. Using $M$ for an ideal is perfectly fine, although if you need it for a module fonts can help. Here it's just unnecessary.
Jul
26
answered solution verification: find characteristic of integral domain under given conditions
Jul
26
awarded  Revival
Jul
25
awarded  Good Answer
Jul
25
answered Modules that have finitely many submodules
Jul
25
comment Modules that have finitely many submodules
A subclass of modules with finitely many submodules would be very tiny, even smaller than the class of Artinian modules over a ring. Even a two dimensional module over the rational numbers has infinitely many submodules. I suppose it would contain the class of simple modules though. These classes are not even closed under finite direct sums...
Jul
25
comment Decompose finitely generated modules and use Krull-Schmidt theorem
Arturo's answer covers this. It's just a matter of pairing up the indecomposables of M, and the the rest necessarily pair up, so their sums are isomorphic.
Jul
24
answered Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$
Jul
24
revised What properties do the rings of infinite, upper-triangular matrices have?
added 376 characters in body
Jul
24
asked What properties do the rings of infinite, upper-triangular matrices have?
Jul
23
comment Idempotent elements of a ring.
@man_in_green_shirt In the example I just gave, $2\Bbb Z_4=(2+4\Bbb Z)$. Probably the source of confusion is the mixing of notations. If you pick one notation you would like to use, I can help you sort it out.
Jul
23
comment Nilpotent minimal ideals are one sided minimal ideals
Oh yes, I have definitely have that paper printed somewhere. Thanks.
Jul
23
comment Idempotent elements of a ring.
@man_in_green_shirt You're mixing coset notations. To write consistently, you would say that $2\Bbb Z_4=\{0+4\Bbb Z, 2+4\Bbb Z\}\lhd \{0+4\Bbb Z,1+4\Bbb Z,2+4\Bbb Z,3+4\Bbb Z\}=\Bbb Z_4$. $2+4\Bbb Z$ is not an ideal at all, it is an element of $\Bbb Z_4$. Surrounding it with parentheses is standard notation for the ideal generated by this element.
Jul
23
comment Nilpotent minimal ideals are one sided minimal ideals
I think you must be still assuming Artinianness in the criterion you mentioned. I am aware of a non Artinian dual ring that satisfies the conditions in the last comment. Do you happen to have a reference for the criterion? I'd like to take a look at it.
Jul
23
comment Nilpotent minimal ideals are one sided minimal ideals
Great! Thanks for adding that. In fact, I am very interested in this area, and I will think about it.
Jul
23
revised Idempotent elements of a ring.
deleted 18 characters in body
Jul
23
comment Idempotent elements of a ring.
@man_in_green_shirt Actually, it looks like the notation is a longstanding typo...
Jul
23
comment Ideals for non commutative ring
"help me for regular elements not for matrix" What does that mean? The word "regular" is used for lots of things in algebra, but I suspect you don't mean any of them, but rather you think that matrices can't be considered as elements for some reason. Matrices over rings are certainly elements of rings.
Jul
23
answered Ideals for non commutative ring
Jul
23
comment Nilpotent minimal ideals are one sided minimal ideals
It's interesting... do you have any more motivation or clues about the problem? Or previous work?