Abstract Algebra Answers

Show that the following conditions on a ring R are equivalent:
(1) All ideals not equal R are prime.
(2) (a) The ideals of R are linearly ordered by inclusion, and (b) All ideals I ⊆ R are idempotent.

For any given division ring k, list all the prime and semiprime ideals in the ring R of 3 × 3 upper triangular matrices over k.

Show that R is prime ring iff it is simple.

Show that in a right artinian ring R, every prime ideal p is maximal.

Show that a ring R is a domain iff R is prime and reduced

Let p ⊂ R be a prime ideal, A be a left ideal and B be a right ideal. Does AB ⊆ p imply that A ⊆ p or B ⊆ p?

For any semiprime ring R, show that Z(R) is reduced, and that char R is either 0 or a square-free integer.

Show that the center Z(R) of prime ring R is an integral domain, and char R is either 0 or a prime number.

Show that a nonzero central element of a prime ring R is not a zero-divisor in R.