Unitary module

A left (or right) module $M$ over a ring with an identity $e$ such that multiplication by $e$ is the identity operator, i.e. the transformation $m\to em$ (respectively, $m\to me$ for right modules), $m\in M$, is the identity automorphism of the group $M$.

Comments

A unitary module as defined above is also (and better) called a unital module, [a1]. The terminology "unitary module" can cause confusion in that it may suggest some module generalization of the concept of a unitary vector space, cf. Unitary space.

Often the property that a module be unital is absorbed into the definition of a module, cf., e.g., [a3].

References

[a1]	P.M. Cohn, "Algebra" , 1–2 , Wiley (1991) pp. 409
[a2]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , v. Nostrand (1958) pp. 134
[a3]	H. Matsumura, "Commutative ring theory" , Cambridge Univ. Press (1989) pp. 7