Question

Does the category of R modules have initial objects? Final objects? (See Exercise 4.14 for the terminology.)

a. True
b. False

Answer 1

The category of R-modules has an initial object but does not have final objects.

Step-by-step explanation:

True, the category of R-modules has an initial object. The initial object in the category of R-modules is the zero module, which consists of the zero element and the zero map from any R-module to the zero module.

On the other hand, the category of R-modules does not have final objects. A final object is an object that every other object maps to, uniquely up to isomorphism. In the category of R-modules, there is no object that every R-module maps to.