Tagged Questions

-6
votes
2answers
59 views

Help with Theorem III.3.11 in Hungerford's algebra book

I need help to prove part (i) of this theorem which I couldn't prove. Any help would be appreciated. Thanks in advance.
3
votes
1answer
34 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
0
votes
1answer
50 views

$M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)

I was rather surprised by the fact that two modules are isomorphic if and only if their abelian group structures are isomorphic. I might just sketch the proof here. Given a ring homomorhpism ...
2
votes
2answers
36 views

Tests/ invariants for module isomorphisms

It two modules are indeed isomorphic, then it is often not too difficult to find an isomorphism since most of the time it is just the natural map. However, it takes some time for me to prove that two ...
1
vote
1answer
116 views

Does every noninvertible element of a commutative ring lie in a proper maximal ideal?

More formally stated: Prove that if $R$ is a commutative ring with $1$, then every element of $R$ that is not invertible is contained in a proper maximal ideal. I know I have to assume Zorn's Lemma, ...
3
votes
2answers
106 views

In general how to prove or disprove certain types of ideal?

i've come across a lot of questions recently that ask you whether or not there exist certain kinds of ideal, say; does there exist an ideal$ J $of $\mathbb{Z}[i]$ for which $\mathbb{Z}[i] /J$ is a ...
1
vote
0answers
60 views

proof that $P^{(n)}$ are primary when $P$ is prime

I am looking for an alternate proof of the fact that in a commutative ring $R$ with a prime ideal $P$, the ideal $P^{(n)}=P^n R_P\cap R$ is primary. I understand once we localize, $P^n R_P$ is a power ...
5
votes
3answers
159 views

Proving that $\mathbb C[C_7]\cong\bigoplus_{i=1}^7\mathbb C$

Let $R$ be the group ring $\mathbb C[C_7],$ where $C_7=\{1,g,\ldots,g^6\}$ is a cyclic group. I would like to prove that $$\mathbb C[C_7]\cong\bigoplus_{i=1}^7\mathbb C.$$ I was thinking that I ...