Abstract Algebra: Rings and Subrings

Recall that if $R$ is a PID and $M$ is a finitely generated $R$-module, then $M$ is isomorphic to the direct sum of cyclic modules whose annihilators are generated by powers of primes in $R$.

Find examples of commutative rings with one that are not integral domains.

Find a PID that is not a Euclidean domain.

Explain the role of irreducible and prime elements in the context of commutative rings with one.

Is a subset S of a ring R a subring of R under the operations of R?

A Boolean ring is any ring where for any element $x$, $x^2 = x$. Prove the following: 1. For all elements $x$ in a Boolean ring, $x + x = 0$. 2. Every Boolean ring is commutative, i.e., for all elements $x$ and $y$ in a Boolean ring, $xy = yx$.

If a ring has a positive characteristic $n$, show that it contains a subring isomorphic to $\mathbb{Z}_n$.