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A torsion abelian group has nonzero socle and is an essential extension of it. Let $R$ be a commutative ring. If $M$ is a unital, torsion $R$-module with nonzero socle, is $M$ an essential extension of its socle?

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Is R an integral domain? Otherwise, I think for an arbitrary torsion theory, the answer is no. – Jack Schmidt Nov 28 '11 at 18:16
@Jack: $R$ could be an integral domain, but I do not want to restrict to that case. In general, I would want to say that $m$ is a torsion element if $rm=0$ for some regular element $r$ in $R$. I don't want to get into an arbitrary torsion theory because I don't understand them all that well yet. – Chris Leary Nov 28 '11 at 19:32
cool. That removes my silly examples ("m is torsion if it is torsion free or annihilated by a power of 2. then Z+Z/2Z has nonzero socle, is torsion, and its socle is not essential"). I'll think about this version. – Jack Schmidt Nov 28 '11 at 21:02
up vote 1 down vote accepted

Here is an example.

Example: Let $R=k[x,y]$ be a polynomial ring in two variables, and let $$M=R/yR \oplus R/(x,y)R = M_1 \oplus M_2.$$ Then y acts as 0 on M, but y is a regular element on R, so M is not only torsion but "bounded". M has socle $M_2$, but $M_1 \cap M_2 = 0$, so the socle $M_2$ is not essential.

Verifying the example is easier if you imagine M as an $S=k[x]$ module, then $M_1 \cong S$ obviously has zero socle as an S-module, and thus as an R-module. $M_2 \cong k$ is simple, so its own socle.

Here is my thought process (removing dead-ends on singular modules). Basically reduce to injectives, then reduce to R/P, and check the condition on R/P. Dim 2 rings are mostly fine, especially ones with lots of primes.

Lemma 1: An essential extension of a torsion-module is torsion.

Lemma 2: Every module has a maximal essential extension, namely its injective envelope.

Lemma 3: Over a commutative noetherian ring, every injective module is a direct sum of injective envelopes of $R/P$ where P is a prime ideal of R.

Lemma 4: A cyclic module $R/I$ is torsion if and only if I contains a regular element.

Lemma 5: A cyclic module $R/I$ has a non-essential socle if I is the intersection of infinitely many prime ideals.

Lemma 6: What we really need is a nonzero non-maximal prime ideal in a noetherian domain.

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Hopefully it is also clear this is the same as my earlier example. Take $R=\mathbb{Z}[x]$, and $y=2$, then M in my answer is suspiciously identical to M in my previous comment. – Jack Schmidt Nov 28 '11 at 22:19
Thanks, Jack. This is exactly the kind of example I was hoping for but could not construct myself. The lemmas will help with a general result I am trying to establish. – Chris Leary Nov 30 '11 at 13:33

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