1
vote
0answers
41 views

Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
0
votes
1answer
22 views

Are the generators of the subgroup defining tensor products linearly independent over $\mathbb Z$?

Let $S$ be a (commutative) ring with identity, and let $M$, $N$ be $S$-modules. (I guess if $S$ isn't commutative, I want $M$ to be a right $S$-module an $N$ a left $S$-module.) In the definition of ...
5
votes
1answer
141 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
2
votes
1answer
123 views

Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
5
votes
0answers
35 views

Groups and Rings [duplicate]

Is every abelian group the additive group of some ring? I would very much appreciate if someone could show me if this is false or true, something I'm thinking about and finding hard to prove so im ...
1
vote
2answers
69 views

Is a set that is an abelian group under addition and a group under multipliation a field?

I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a ...
0
votes
2answers
31 views

Isomorphism of rings under different operations

i am new to this site. i have been reading through its posts and question and they are really amazing. however, i found a link to a question asked about one year ago and a question i don't know how to ...
4
votes
1answer
56 views

Extending abelian groups to rings

I've been reading this article about extending abelian groups to rings: http://www.math.udel.edu/~coulter/papers/rings.pdf. Could you explain to me why theorem 2.1 guarantees left and right ...
-1
votes
1answer
189 views

How to turn a group ring $R(G)$ into a ring?

Let $R(G)$ be a given abelian group ring. Any abelian group ring is isomorphic to an abelian ring. I know how to express (isomorphism) some group rings as a ring. But I wonder if there is a general ...
1
vote
1answer
59 views

Fields arising as endomorphism rings

Do you know a field $K$ other than $F_p$ which is the endomorphism ring of an abelian group $G$? I doubt that there is one because as $G$ gets bigger, $End(G)$ seems to be more and more ...
8
votes
2answers
303 views

Does every ring with unity arise as an endomorphism ring?

I don't believe that every ring with a $1$ is the endomorphism ring of an abelian group but I currently don't see how to produce a counterexample.
2
votes
2answers
284 views

Prove that R is a ring under 'special' definitions of multiplication and addition

Question: Let R be a ring with a 1. Define $\bar R$ to have the same elements of R with addition $$\oplus: a \oplus b = a +b +1$$ andmultiplication $$\otimes: a \otimes b = ab + a +b$$ Prove that ...
0
votes
2answers
98 views

Relation between torsion subgroup of multiplicative group of field and solvability of polynomials

In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have ...
2
votes
0answers
170 views

A question about reduced torsion abelian groups

If a reduced torsion abelian group has no cyclic direct summands of order greater than 2, is it an elementary abelian 2-group? Background: I'm trying to classify the groups whose group rings have a ...