Exact embedding functors and left coherent rings

Authors:
Kent R. Fuller and George Hutchinson

Journal:
Proc. Amer. Math. Soc. **104** (1988), 385-391

MSC:
Primary 16A89; Secondary 18E20

MathSciNet review:
962803

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Abstract: Let and be rings with unit. Suppose is a free -module on generators, where is an infinite cardinal number not smaller than the cardinality of , and is the ring of endomorphisms .

Theorem. *If* *is left coherent and there exists an exact embedding functor* , *then* *is a bimodule such that* *is faithfully flat*.

Theorem. *If* *is an exact embedding functor, then* *is a bimodule such that* *is a projective generator (inducing an exact embedding* Hom *functor from* *into* ,) *and* *is a bimodule such that* *is faithfully flat (inducing an exact embedding tensor product functor* -- *from* *into* .)

Theorem. *There exists an exact embedding functor* *iff there exists an* *-module* *and a unit-preserving ring monomorphism* *of their endomorphism rings, such that* *preserves and reflects exact pairs of endomorphisms*.

**[1]**Frank W. Anderson and Kent R. Fuller,*Rings and categories of modules*, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, Vol. 13. MR**0417223****[2]**Stephen U. Chase,*Direct products of modules*, Trans. Amer. Math. Soc.**97**(1960), 457–473. MR**0120260**, 10.1090/S0002-9947-1960-0120260-3**[3]**Peter Freyd,*Abelian categories. An introduction to the theory of functors*, Harper’s Series in Modern Mathematics, Harper & Row, Publishers, New York, 1964. MR**0166240****[4]**Kent R. Fuller,*Density and equivalence*, J. Algebra**29**(1974), 528–550. MR**0374192****[5]**George Hutchinson,*Exact embedding functors between categories of modules*, J. Pure Appl. Algebra**25**(1982), no. 1, 107–111. MR**660390**, 10.1016/0022-4049(82)90095-0**[6]**George Hutchinson,*Addendum to: “Exact embedding functors between categories of modules” [J. Pure Appl. Algebra 25 (1982), no. 1, 107–111; MR0660390 (83k:16030)]*, J. Pure Appl. Algebra**45**(1987), no. 1, 99–100. MR**884634**, 10.1016/0022-4049(87)90087-9**[7]**Barry Mitchell,*Theory of categories*, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR**0202787****[8]**Charles E. Watts,*Intrinsic characterizations of some additive functors*, Proc. Amer. Math. Soc.**11**(1960), 5–8. MR**0118757**, 10.1090/S0002-9939-1960-0118757-0

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0962803-9

Article copyright:
© Copyright 1988
American Mathematical Society