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I have seen the term "contre-module" in Bourbaki's "Algèbre, Chapitre VIII" in the first few pages. The notion sounds useful in the theory of non-commutative rings, but I couldn't google an equivalent in English. Two questions :

  • Is this notion commonly used by non-commutative algebraists?
  • Does it have a fairly commonly used English equivalent?

Definition : Let $A$ be a (non-commutative unital) ring and $M$ a left $A$-module. The "contre-module" of $M$ is the $\mathrm{End}_A(M)$-module $M$ whose underlying abelian group is equal to $M$ and multiplication is given by $$ \forall \varphi \in \mathrm{End}_A(M), \quad \forall m \in M, \quad \varphi \cdot m \overset{def}= \varphi(m). $$

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    Can you include the definition they give, so that people who do not have the book at hand know what the question is about? – 2017-02-27
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    @MarianoSuárez-Álvarez : Added, good point. – 2017-02-27
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    Googling for "counter module" turns up various things that seem pertinent. – 2017-02-27
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    (To he flagger: I prefer to keep my comment above, so that everyone reading the question is reminded of the fact that we have to write questions in a gnerally useful way as possible!) – 2017-02-27
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    I don't recall seeing a name for that module, but certainly it is used. People introduce names for properties of the contre-module without naming it explicitly: for example, algebra-representation people talk about *endofinite modules*, which are modules over a ring which are of finite length when viewed as modules over their endomorphism ring, and so on. – 2017-02-27
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    @MarianoSuárez-Álvarez "I prefer to keep my comment above" And I'd prefer you had left that decision to a different moderator. It's a small thing but still poor style in my opinion. Such comments are routinely deleted. – 2017-02-27
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    Indeed. And as they are generally deleted, we are constantly in the need of remaking them. – 2017-02-27
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    @MarianoSuárez-Álvarez yeah, right, as can be seen on MO where basically nothing ever got deleted the need to make such comments disappeared over time. Except well it did not. Anyway, now the comment indicating that the word "counter module" might be used needs expansion to be seen... surely that's less relevant than yours. – 2017-02-27

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