## Math 120B HW 1

Due Tuesday, April 7.
- For each part, say whether or not the given structure is a ring. Justify your answers.
- $(\mathbb{Z},+,\max)$ where $\max$ denotes the operation that takes the maximum of two given integers.
- $(\mathbb{Z},+,\ast)$ where $a \ast b = 0$ for all $a,b \in \mathbb{Z}$.
- $(\mathbb{Z},\ast,\ast)$ where $\ast$ denotes ordinary addition (so ring addition and ring "multiplication" are both ordinary addition.)

- Let $n \in \mathbb{Z}^+$. Recall that $\mathbb{Z}_n$ denotes the ring with elements $\{0,\ldots,n-1\}$ under the operations of addition modulo $n$ and multiplication modulo $n$.
- Assume that $n$ is prime. Prove that for all elements $a,b \in \mathbb{Z}_n$, if $ab = 0$ then $a = 0$ or $b = 0$.
- Assume that $n$ is composite. Prove that there are some elements $a,b \in \mathbb{Z}_n$ such that $ab = 0$ but $a \ne 0$ and $b \ne 0$.

- For each part, say whether or not the given function is a homomorphism (of rings.) Justify your answers.
- The function $\phi : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ defined by $\phi((a,b)) = a+b$.
- The function $\phi : M_2(\mathbb{R}) \to \mathbb{R}$ defined by $\phi(A) = \det(A)$ (the determinant function.)
- The function $\phi : \mathbb{Q} \to \mathbb{Z}$ defined by $\phi(a/b) = a$.

* Warning: one of these "functions" is not well-defined. So it is a trick question. *
- Let $\phi:\mathbb{Z} \to \mathbb{Z}$ be a homomorphism (of rings.) Prove that $\phi$ is either trivial (maps every element to $0$) or the identity (maps every element to itself.)
*Hint: if $a,b \in \mathbb{Z}$ then the product $ab$ can be interpreted in two ways; either by multiplication in the ring or by repeated addition in the ring. So if you apply $\phi$ to $ab$ you can use the definition of "homomorphism" in two different ways.*
- Let $(F,+,\cdot)$ be a field and let $F^*$ denote the set of all nonzero elements of $F$. Prove that $(F^*, \cdot)$ is an abelian group. (It is called the
*multiplicative group* of the field $F$.) *Hint: don't forget to prove the relevant closure properties.*
- Find the multiplicative inverse of each nonzero element of $\mathbb{Z}_{11}$. Which elements of $\mathbb{Z}_{12}$ have multiplicative inverses? Find these inverses.