## Math 120B HW 2

Due Tuesday, April 14.
1. An element $a$ of a ring is called idempotent if $a^2 = a$. In any ring, $0$ is idempotent, and $1$ is idempotent if it exists.
1. Prove that if $D$ is an integral domain, then $0$ and $1$ are the only idempotent elements of $D$.
2. Find all idempotent elements of the ring $\mathbb{Z}_{15}$.
3. Recall that the function $f : \mathbb{Z}_{15} \to \mathbb{Z}_3 \times \mathbb{Z}_5$ defined by $f(a) = (a \bmod 3, a \bmod 5)$ is an isomorphism because $15 = 3 \times 5$ and $\text{gcd}(3,5) = 1$. Explain your answer to part b in terms of this isomorphism.
2. Let $R$ and $R'$ be commutative rings with unity $1 \ne 0$.
1. Prove that if $R$ is an integral domain, and $R'$ is isomorphic to $R$, then $R'$ is an integral domain.
2. Give an example where $R$ is an integral domain, $R'$ is not an integral domain, and there is a surjective homomorphism $\phi: R \to R'$.
3. Give an example where $R$ is not an integral domain, $R'$ is an integral domain, and there is a surjective homomorphism $\phi : R \to R'$.
3. Let $n \in \mathbb{Z}^+$. Prove that every nonzero element of the ring $M_n(\mathbb{R})$ is either a unit or a zero divisor. (You will need to remember some linear algebra for this problem.)
4. Let $R$ be a commutative ring with unity $1 \ne 0$ and let $a \in R$. Define a function $\ell_a : R \to R$ by $\ell_a(b) = ab$.
1. Prove that $\ell_a$ is injective if and only if $a \ne 0$ and $a$ is not a zero divisor.
2. Prove that $\ell_a$ is surjective if and only if $a$ is a unit.
5. Let $R$ be a ring with unity $1_R$ and let $S$ be a subring of $R$ with unity $1_S$.
1. Prove that if $1_R \in S$ then the characteristic of $S$ equals the characteristic of $R$.
2. Find an example where $1_R \notin S$ and the characteristic of $S$ is not equal to the characteristic of $R$.
6. Prove the following statement, or find a counterexample: "if $R$ is a ring with unity and the characteristic of $R$ is prime, then $R$ is an integral domain."