Basics of Rings

Prove that $\mathbb{Z}$ is a ring under regular addition and multiplication. You do not need to prove the Distributive axiom.

Does the ring $\mathcal{M}_{2\times 2}(\mathbb{R})$ have zero divisors?

Which of the following rings, if any, are fields?

1. $\mathbb{Z}$

2. $\mathbb{Q}$

Prove 1.30.

Show that

is a subring of $\mathcal{M}_{2\times 2}(\mathbb{R})$.

Let $R$ be a ring. Prove that if $u\in R$ is a unit then so is $-u$.

Prove that the trivial ring is the unique ring with identity in which $0=1$.

Let $R$ be a set with an operation $\ast$ such that for all $x\ast y\in R$ for all $x,y\in R$. If $(R,\ast,\ast)$ is a ring, describe the elements in $R$.

Let $R$ be a ring with identity 1, and let $x\in R$.

1. Find four closed forms for the geometric series

2. What are the condition(s) such that each of the closed forms are valid?

3. Factor 112 into primes, and thus evaluate 112 in $\mathbb{Z}_{37}$.

4. Hence, using the result(s) above, evaluate

   in the ring $\mathbb{Z}_{37}$.

Show that

is a ring. Hence show it is a field.

Let

be a ring under matrix addition and multiplication.

1. Show that $R$ has no identity.

2. Show that $R$ contains a non-trivial subring $S$ with identity.

A ring $R$ is called a Boolean ring if $r^{2}=r$ for all $r\in R$.

1. Show that $r=-r$ for all $r\in R$.

2. Prove that every Boolean ring is commutative.

Let $R$ be a commutative ring with identity. We say that an element $x\in R$ is nilpotent if there exists a positive integer $n$ such that $x^{n}=0$.

1. Let $u\in R$ be a unit and $x\in R$ be nilpotent. Show that $ux$ is nilpotent.

2. Show that $u-x$ is a unit.