• MHB
• steenis
In summary, this conversation is about an example in Rotman's book "Advanced Modern Algebra" 2nd edition 2010, page 404. Let $M$ be a left $R$-module over a ring $R$, let $J$ be a left ideal in $R$ generated by $r$, and let $JM$ be a submodule of $M$. Prove that $JM$ is a subgroup of $M$.
steenis
Please, can someone help me with this?

Let $M$ be a left $R$-module over a ring $R$.
Let $J$ be a left ideal in $R$ generated by $r$: $J=Rr=<r>$.
Now $JM=\{am \ | \ a \in J \ and \ m \in M\}$
Prove that $JM$ is a submodule of $M$.

This is an example in Rotman's book "Advanced Modern Algebra" 2nd edition 2010, page 404.

steenis said:
Please, can someone help me with this?

Let $M$ be a left $R$-module over a ring $R$.
Let $J$ be a left ideal in $R$ generated by $r$: $J=Rr=<r>$.
Now $JM=\{am \ | \ a \in J \ and \ m \in M\}$
Prove that $JM$ is a submodule of $M$.

This is an example in Rotman's book "Advanced Modern Algebra" 2nd edition 2010, page 404.

Hi steenis,

From wiki:
Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and any r in R, the product r ⋅ n is in N.

So don't we need to verify that $JM$ is a subgroup of $M$, and that for any $a_1m_1$ in $JM$ and any $\tilde r$ in $R$, the product $\tilde r a_1 m_1$ is in $JM$, knowing that $a_1 = r_1r$?

Yes, I thought so too. But how do I prove that if $x,y \in JM$ then $x+y \in JM$ ?.
That is $x=r_1rm$ and $y=r_2rn$, then $r_1rm+r_2rn$ has to be in $JM$.
I feel stupid but I do not see it.

It is true if $r \in Z(R)$, the center of $R$. Howerver Rotman explicitly stated "If $r$ is an element of $R$ not in the center of $R$ $...$".

The set $JM$ should consist of finite sums of the form $am$ where $a\in J$ and $m\in M$, otherwise closure under addition fails. For example, let $R$ be the ring of $2\times 2$ real matrices, and let $M = R$, viewed as a left $R$-module. If $r = \begin{bmatrix}1&0\\0&0\end{bmatrix}$, then $J = \left\{\begin{bmatrix}a&0\\c&0\end{bmatrix}:a,c\in \Bbb R\right\}$, and so $JM = \left\{\begin{bmatrix}ax&ay\\cx&cy\end{bmatrix}: a,c,x,y\in \Bbb R\right\}$. In particular, elements of $JM$ have zero determinant. This set is not a submodule of $M$, for both $\begin{bmatrix}1&0\\0&0\end{bmatrix}$ and $\begin{bmatrix}0&0\\0&1\end{bmatrix}$ are in $JM$, but the sum, $\begin{bmatrix}1&0\\0&1\end{bmatrix}$, is not a member of $JM$ since it has nonzero determinant.

I actually tried sending an email to him about Example 6.18(v) but received an email (from his daughter, I believe) stating that he passed away 18 months ago. :(

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That is sad news, I did not know that. I saw on the internet that he passed away on 16 october 2016, that is 83 years old.

Thank you for looking at my problem, but I do not understand your example, because if $R$ is the ring of 2x2 real matrices and $r$ is the identity, then $J=R$. Can you explain your example once more?

That was a typo; the second row was meant to be a row of zeros. I'm fixing some of the other typos.

The typos are now fixed.

Now I understand, thank you.

Thank you for trying to contact Rotman.

Strangely, this example still appears in the next edition of his book (B-1.26(vi)).

Last edited:
Funny you mentioned that -- looking back at the section, I see that he used nearly the same 'example' as my counterexample. (Giggle)

1. What is a submodule in advanced modern algebra?

A submodule is a subset of a module that is closed under addition and scalar multiplication by elements of the underlying ring. In other words, it is a subset of a module that is itself a module over the same ring.

2. What is the main problem discussed in "Advanced Modern Algebra" by Rotman?

The main problem discussed in this book is the classification of finitely generated modules over a principal ideal domain (PID). This problem is known as the structure theorem for finitely generated modules over a PID.

3. How does the structure theorem for finitely generated modules over a PID relate to submodules?

The structure theorem states that every finitely generated module over a PID can be decomposed into a direct sum of cyclic submodules, which are generated by a single element. This is useful in determining the structure of submodules within a larger module over a PID.

4. What are some applications of the structure theorem for finitely generated modules over a PID?

One important application is in the study of linear algebra, where the structure theorem allows us to classify and understand the structure of vector spaces over a field. It is also used in algebraic number theory to study number fields and their ring of integers.

5. Are there any other important results or theorems related to submodules in advanced modern algebra?

Yes, there are many other important results and theorems related to submodules, such as the lattice isomorphism theorem, Nakayama's lemma, and the Artin-Wedderburn theorem. These results are fundamental in the study of module theory and have many applications in other areas of mathematics.

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