Any field that is finitely generated as a ring is a finite field.
Exhibit a group , a set of generators for and a subgroup such that for all and yet is not normal in .
Is there a ring such that there are two different ring injections ?
Let be a ring epimorphism. Suppose is an -linear mapping between -modules, where the -linear structure is given by . Does it follow that is -linear?
Let be a commutative ring, ideals. If as -modules, then .
Are there any decomposable quotients of (viewed as a -module)?
Suppose is a -graded connected algebra over a field concentrated in positive degrees, and suppose is a projective graded -module concentrated in degrees above some . Then is actually a free -module.
Under the same hypothesis over , show that if is a graded module and where is the quotient of by the ideal of positively graded elements, then .
Using the above, show that if is graded and , then is free. To illustrate, it follows projective graded modules over polynomial algebras are free.