*Categorical epimorphisms and monomorphisms.*A morphism h : A -----> B in a category**Gamma**is an**epimorphism**if for each object X in**Gamma**and each pair of morphisms j : B -----> X and k : B -----> X one has j \circ h = k \circ h if and only if j = k. A**monomorphism**in**Gamma**is an epimorphism in the dual category**Gamma**^{op} (with all arrows reversed).Identify the monomorphisms and epimorphisms in the category of modules over a given ring.

Show that a monomorphism in the category of rings is the same thing as an injective ring homomorphism.

Show that every surjective ring homomorphism is an epimorphism of rings but that not every epimorphism of rings is surjective.

*Hint:*Look for an example in the previous assignment.

Let M, N, and P be R-modules, and suppose that there are given R-linear maps

p : P -----> M q : P -----> N i : M -----> P j : N -----> P that satisfy the relations (1) p \circ i = 1, (2) q \circ j = 1, (3) p \circ j = 0, (4) q \circ i = 0, and (5) i \circ p + j \circ q = 1 (this last in Hom_{R}(P, P)). Show that the triple (P, p, q) is a product in the category of R-modules.

Let R be a given ring, A a given R-module, and let

**Gamma**be the category of pairs (X, f) where X is an R-module and f : X -----> A is an R-linear map. Construct a product in the category**Gamma**for two given objects (M, g) and (N, h).*Hint:*It may be helpful first to do the analogous exercise when the category of R-modules is replaced by the category of sets.Compute the following tensor products:

R \otimes_{R} M when R is a ring and M an R-module.

(

**Z**/m**Z**) \otimes_{**Z**} (**Z**/n**Z**) for m, n >= 1.**C**\otimes_{**R**} M_{n}(**R**) where**R**is the real field,**C**the complex field, and M_{n}() the functor for forming the ring of n \times n matrices.

*Hint:*A technique for doing such examples is first to guess the correct answer, possibly after examining special cases and then to show that the guessed answer satisfies the universal mapping property which characterizes the given object.Let R be a ring and M, N 2-sided R-modules. Under these conditions one certainly has, as discussed in class, an abelian group T = M \otimes_{R} N. Under what circumstances may T be given the structure of 2-sided R-module?

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