Dear colleagues, A semiring is roughly speaking a ring without subtraction, i.e. (R,+,0) is an Abelian monoid & (R,*,1) is a semigroup with distribution of * over + (e.g. the set of non-negative integers). A semimodule over a semiring is roughly speaking a module without subtraction, i.e. (M,+,0) is an Abelian Monoid, and there is a scalar multiplication of the semiring on M with the usual expected properties. A semimodule over a semiring is cancellative, if the Abelian monoid (R,+) is cancellative. The category of N0-Semimoduels is just the category of "Commutative Monoids". Indeed the category of left semimodules over an arbitrary semiring R? (A special example would be the category of commutative monoids) is not pre-additive. However, for any left semimodules M and N over a semiring R, (Hom_R(M,N),+,0) is an Abelian monoid and it has kernels and cokernels. A monomorphism of semimodules is injective, however only an "image-regular" epimorphism is subjective. For a morphism of semimodules f: M --> N, the sequence 0 ---> Coimage(f) --> Im(f) --> is exact, but the canonical map Coimage(f) --> Im(f) is not an isomorphism (neither a bimorphism), unless f is regular. The category of semimodules had products, equalizers and products (however not necessarily coequalizers). The category of cancellative semimodules is complete and cocomplete. It has a generator, namely the semiring itself and I also "expect" it to have exact colimits (ANY REFERENCE?). What kind of Categories is the category of (cancellative) semimodules over semiring? Is there a notion of "almost Grothendieck categories" or "Semi-Grothendieck Categories" to which categories of (cancellative) semimodules fit? Unfortunately, I did not find a single book that clarifies the categorical aspects of semimodules (The 3 books of Golan as well as the books on the subject by others are devoted more to semirings and automata and not have much on semimodules). I appreciate very much your comments, suggestions for literature (e.g. books, Ph.D. dissertations, articles) on the CATEGROY OF (Cancellative) SEMIMODULES (other than the papers of Takahashi and Katsov, which I already have). With best regards, Jawad ----------------------------------------------------- Dr. Jawad Abuhlail Dept. of Math. & Stat. Box # 5046, KFUPM 31261 Dhahran (KSA) http://faculty.kfupm.edu.sa/math/abuhlail