Chapter 17

In a ring with unity, prove that if \(a\) is nilpotent, then \(a+1\) and \(a-1\) are both invertible. [HINT: Use the factorization $$ 1-a^{n}=(1-a)\left(1+a+a^{2}+\cdots+a^{n-1}\right) $$ for \(1-a\), and a similar formula for \(1+a\).]

A ring \(A\) is a boolean ring if \(a^{2}=a\) for every \(a \in A\). Prove that each of the following is true in an arbitrary boolean ring \(A\). For every \(a \in A, a=-a\). [HINT: Expand \((a+a)^{2}\).]

Prove that each of the following is true in a nontrivial ring with unity. If \(a \neq \pm 1\) and \(a^{2}=1\), then \(a+1\) and \(a-1\) are divisors of zero.

Prove In any ring, \(a(b-c)=a b-a c\) and \((b-c) a \pm b a-c a\).

If \(A\) and \(B\) are rings, verify that \(A \times B\) is a ring.

Verify that \(\mathscr{M}_{2}(\mathrm{R})\) satisfies the ring axioms.

In each of the following, a set \(A\) with operations of addition and multiplication is given. Prove that \(A\) satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity, and the negative of an arbitrary \(a\). \(A\) is the set \(\mathbb{Z}\) of the integers, with the following "addition" \(\oplus\) and "multiplication" \(\odot\) : $$ a \oplus b=a+b-1 \quad a \odot b=a b-(a+b)+2 $$

Prove that each of the following is true in a nontrivial ring with unity. An element \(a\) can have no more than one multiplicative inverse.

Prove In any ring, if \(a b=-b a\), then \((a+b)^{2}=(a-b)^{2}=a^{2}+b^{2}\).

In each of the following, a set \(A\) with operations of addition and multiplication is given. Prove that \(A\) satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity, and the negative of an arbitrary \(a\). \(A\) is the set \(\mathbb{Q}\) of the rational numbers, and the operations are \(\oplus\) and \(\odot\) defined as follows: $$ a \oplus b=a+b+1 \quad a \odot b=a b+a+b $$

Prove that each of the following is true in a nontrivial ring with unity. In a commutative ring with unity, a divisor of zero cannot be invertible.

Prove In any integral domain, if \(a^{2}=b^{2}\), then \(a=\pm b\).

Explain why \(\mathscr{M}_{2}(\mathbb{R})\) is not an integral domain or a field.

In each of the following, a set \(A\) with operations of addition and multiplication is given. Prove that \(A\) satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity, and the negative of an arbitrary \(a\). \(A\) is the set \(\mathbb{Q} \times \mathbb{Q}\) of ordered pairs of rational numbers, and the operations are the following addition \(\oplus\) and multiplication \(\odot:\) $$ \begin{aligned} &(a, b) \oplus(c, d)=(a+c, b+d) \\ &(a, b) \odot(c, d)=(a c-b d, a d+b c) \end{aligned} $$

In a commutative ring, prove that the product of two unipotent elements \(a\) and \(b\) is unipotent. [HINT: Use the binomial formula to expand \(1-a b=\) \((1+a)+a(1-b)\) to power \(n+m .]\)

Prove that each of the following is true in a nontrivial ring with unity. Suppose \(a b \neq 0\) in a commutative ring. If either \(a\) or \(b\) is a divisor of zero, so is \(a b\).

In any integral domain, only 1 and \(-1\) are their own multiplicative inverses. (Note that \(x=x^{-1}\) iff \(x^{2}=1\).)

The conjugate of \(\alpha\) is $$ \bar{\alpha}=\left(\begin{array}{lr} a-b i & -c-d i \\ c-d i & a+b i \end{array}\right) $$ The norm of \(\alpha\) is \(a^{2}+b^{2}+c^{2}+d^{2}\), and is written \(\|\alpha\| .\) Show directly (by matrix multiplication) that $$ \bar{\alpha} \alpha=\alpha \bar{\alpha}=\left(\begin{array}{cc} t & 0 \\ 0 & t \end{array}\right) \quad \text { where } t=\|\alpha\| $$ Conclude that the multiplicative inverse of \(\alpha\) is \((1 / t) \vec{\alpha}\).

In each of the following, a set \(A\) with operations of addition and multiplication is given. Prove that \(A\) satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity, and the negative of an arbitrary \(a\). \(A=\\{x+y \sqrt{2}: x, y \in \mathbb{Z}\\}\) with conventional addition and multiplication.

In a commutative ring, prove that every unipotent element is invertible. (HINT: Use the binomial expansion formula.)

Assume \(A\) has a unity. Letting \(a \vee b=a+b+a b\), we have the following in \(A\) : $$ a \vee b c=(a \vee b)(a \vee c) \quad a \vee(1+a)=1 \quad a \vee a=a \quad a(a \vee b)=a $$

Prove that each of the following is true in a nontrivial ring with unity. Suppose \(a\) is neither 0 nor a divisor of zero. If \(a b=a c\), then \(b=c\).

The set \(S\) of all the invertible elements in a ring is a multiplicative group.

Show that the commutative law for addition need not be assumed in defining a ring with unity: it may be proved from the other axioms. [HINT: Use the distributive law to expand \((a+b)(1+1)\) in two different ways.]

Prove that each of the following is true in a nontrivial ring with unity. \(A \times B\) always has divisors of zero.

By part 5, the set of all the nonzero elements in a field is a multiplicative group. Now use Lagrange's theorem to prove that in a finite field with \(m\) elements, \(x^{m-1}=1\) for every \(x \neq 0\)

Let \(A\) be any ring. Prove that if the additive group of \(A\) is cyclic, then \(A\) is a commutative ring

If \(a x=1, x\) is a right inverse of \(a\); if \(y a=1, y\) is a left inverse of \(a\). Prove that if \(a\) has a right inverse \(x\) and a left inverse \(y\), then \(a\) is invertible, and its inverse is equal to \(x\) and to \(y\). (First show that \(y a x a=1\).)

In any integral domain, if \(a^{n}=0\) for some integer \(n\), then \(a=0\).