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If $M$ is an $R$-module and $M_1, M_2$ are submodules of $M$, then one can construct the ideal $\{ r \in R \mid rM_2 \subseteq M_1 \}$, which is denoted $(M_1 : M_2)$. Does this construction have a name?

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The notation probably originates from the ideal quotient, but I've never heard a name for this extension. – Dylan Moreland Jun 15 '12 at 4:56
I would probably call it an annihilator if I had to choose a name – DBr Jun 15 '12 at 7:00
up vote 2 down vote accepted

I would call it the residual $(M_1:M_2)$ because of some papers I encountered by Ward, Dilworth and others that were oriented around residuated lattices.

To keep the story short, I'll just say that they were abstracting commutative ring ideas out to the study of the lattice of ideals. The residual can be used to make a residuated mapping on the lattice of ideals of a ring.

Residuals pop up with different terminology and in tangential ideas:

  • If $M_1$ and $M_2$ are ideals in a ring, then it is also called the ideal quotient. I have also seen transporter and conductor applied, even when $M_1$ is merely a set.

  • When $M_1=\{0\}$ this is just another way of writing the annihilator of $M_2$. In fact you can think of it this way in general: $(M_1:M_2)=\mathrm{ann}((M_2+M_1)/M_1)$.

  • When $M_1$ is a right ideal of a ring, then $(M_1:M_1)$ (as you have defined it) is the idealizer of $M_1$. It is the largest subring of $R$ in which $M_1$ is a two-sided ideal.

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Though if we call $(I:J)$ ideal quotient I wonder why we don't call $(M:N)$ a module quotient. – Rudy the Reindeer Jun 15 '12 at 11:46
@MattN. I haven't encountered it that way, but that would be a good name for it. "Residual" is really the way lattice theorists think of it. – rschwieb Jun 15 '12 at 11:50
@MattN. My worry is that forming the quotient or factor module out of modules $M_1 \subset M_2$ seems deserving of that name as well. – Dylan Moreland Jun 15 '12 at 18:34
@DylanMoreland Indeed. OTOH, aren't there other cases where one word means two completely different things? – Rudy the Reindeer Jun 15 '12 at 19:14
@MattN. Numerous cases, but we don't need to be adding fuel to the fire if we can help it :) – rschwieb Jun 15 '12 at 19:26

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