Coherent ring

A ring in which every finitely-generated left ideal is finitely presentable, that is, it is the quotient module of a finitely-generated free module by a submodule which is an image of a finitely-generated free module. Such a ring is called a left coherent ring; a right coherent ring is defined similarly in terms of right ideals. A left coherent ring can also be defined by either of the following two equivalent conditions: 1) every finitely-generated submodule of a finitely-presented left $ R $- module is finitely presentable; 2) a direct product of flat right $ R $- modules is a flat right $ R $- module.

Many constructions known for modules over Noetherian rings (cf. Noetherian ring) prove to be realizable for modules over coherent rings as well. For example, every finitely-presented module over a coherent ring has a projective resolution by finitely-presented modules. At the same time, the class of coherent rings is wider than that of Noetherian rings since it contains, for example, all regular rings (in the sense of von Neumann, cf. Regular ring (in the sense of von Neumann)) and the rings of polynomials over Noetherian rings in any (finite or infinite) number of variables.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) pp. Chapt. 1 (Translated from French)

Comments

A left $ A $ module $ E $ is finitely presentable if there is an exact sequence

$$ A ^ {r} \rightarrow A ^ {s} \rightarrow E \rightarrow 0 ,\ \ r , s \in \mathbf N . $$

A left $ A $- module is pseudo-coherent if every submodule of finite type (i.e. with a finite number of generators) is finitely presentable, and it is a coherent module if it is also itself of finite type. A left module $ E $ is coherent if and only if for every morphism of left modules $ f : F \rightarrow E $ with $ F $ of finite type the kernel of $ f $ is also of finite type. Thus, $ A $ is a left coherent ring if $ A $, considered as a left $ A $- module, is a coherent left $ A $- module. This is equivalent to the requirement that every finitely-presented left module be coherent.

More generally, in suitable Abelian categories one defines a coherent object as an object $ X $ of finite type such that for any morphism $ f : Y \rightarrow X $ with $ Y $ of finite type, $ \mathop{\rm Ker} f $ is also of finite type, [a1].

References

[a1]	N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973)