Chapter 3.3, Problem 15E

We define a subring of a ring in the same way we defined a subgroup of a group: $\left(S,+,\cdot \right)$ is a subring of the ring $\left(R,+,\cdot \right)$ if $S\subseteq R,$ and $\left(S,+,\cdot \right)$ is a ring with the same operations. For example, the ring of even integers is a subring of the ring of integers, and both are subrings of the ring of rational numbers.

1. Prove that the ring $\left(\left\{0\right\},+,\cdot \right)$ is a subring of any ring $\left(R,+,\cdot \right)$ (called the trivial subring).
2. (Subring Test) Prove that if $\left(R,+,\cdot \right)$ is a ring, Tis a nonempty subset of R, and T is closed under subtraction and multiplication, then $\left(T,+,\cdot \right)$ is a subring.