Chapter 18

Let \(A\) be a ring, and let \(J\) and \(K\) be ideals of \(A\). Prove each of the following. If \(J \cap K=\\{0\\}\), then \(j k=0\) for every \(j \in J\) and \(k \in K\).

Let \(A\) and \(B\) be rings, and \(f: A \rightarrow B\) a homomorphism. Prove each of the following. \(f(A)=\\{f(x): x \in A\\}\) is a subring of \(B\)

Prove that each of the following is a homomorphism. Then describe its kernel and its range. \(\phi: \mathscr{F}(\mathbb{R}) \rightarrow \mathbb{R}\) given by \(\phi(f)=f(0)\)

A nonempty subset \(B\) of a ring \(A\) is closed with respect to addition and negatives iff \(B\) is closed with respect to subtraction.

Identify which of the following are ideals of \(\mathbb{Z} \times \mathbb{Z}\), and explain: \(\\{(n, n): n \in \mathbb{Z}\\}\) \(\\{(5 n, 0): n \in \mathbb{Z}\\} ;\\{(n, m): n+m\) is even \(\\} ;\\{(n, m): n m\) is even \(\\} ;\\{(2 n, 3 m): n, m \in \mathbb{Z}\\}\)

Prove that each of the following is a subring of the indicated ring. \(\\{x+\sqrt{3} y: x, y \in \mathbb{Z}\\}\) is a subring of \(\mathbb{R}\)

Let \(A\) be a ring, and let \(J\) and \(K\) be ideals of \(A\). Prove each of the following. For any \(a \in A, I_{a}=\\{a x+j+k: x \in A, j \in J, k \in K\\}\) is an ideal of \(A\).

Let \(\mathscr{S}\) be the following subset of \(\mathscr{M}_{2}(\mathbb{R})\) : $$ \mathscr{S}=\left\\{\left(\begin{array}{rr} a & b \\ -b & a \end{array}\right): a, b \in \mathbb{R}\right\\} $$ Prove that the function $$ f(a+b \mathbf{i})=\left(\begin{array}{rr} a & b \\ -b & a \end{array}\right) $$ is an isomorphism from \(\mathbb{C}\) to \(\mathscr{S}\). [REMARK: You must begin by checking that \(f\) is a well-defined function; that is, if \(a+b \mathbf{i}=c+d \mathbf{i}\), then \(f(a+b \mathbf{i})=f(c+d \mathbf{i})\). To do this, note that if \(a+b \mathbf{i}=c+d \mathbf{i}\) then \(a-c=(d-b) \mathbf{i} ;\) this last equation is impossible unless both sides are equal to zero, for otherwise it would assert that a given real number is equal to an imaginary number.].

Let \(A\) and \(B\) be rings, and \(f: A \rightarrow B\) a homomorphism. Prove each of the following. The kernel of \(f\) is an ideal of \(A\).

Prove that each of the following is a homomorphism. Then describe its kernel and its range. \(h: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) given by \(h(x, y)=x\)

If \(A\) is a ring with unity, prove that \(J\) is an ideal of \(A\) iff \(J\) is closed with respect to addition and \(J\) absorbs products in \(A\).

List all the ideals of \(\mathbb{Z}_{12}\)

Let \(A\) be a ring, and let \(J\) and \(K\) be ideals of \(A\). Prove each of the following. The radical of \(J\) is the set rad \(J=\left\\{a \in A: a^{n} \in J\right.\) for some \(\left.n \in \mathbb{Z}\right\\} .\) For any ideal \(J, \operatorname{rad} J\) is an ideal of \(A\).

Prove that \(\\{(x, x): \mathrm{x} \in \mathbb{Z}\\}\) is a subring of \(\mathbb{Z} \times \mathbb{Z}\), and show \(\\{(x, x): x \in \mathbb{Z}\\} \cong \mathbb{Z}\).

Let \(A\) and \(B\) be rings, and \(f: A \rightarrow B\) a homomorphism. Prove each of the following. \(f(0)=0\), and for every \(a \in A, f(-a)=-f(a)\)

Prove that each of the following is a homomorphism. Then describe its kernel and its range. \(h: \mathbb{R} \rightarrow \mathscr{M}_{2}(\mathbb{R})\) given by $$ h(x)=\left(\begin{array}{ll} x & 0 \\ 0 & 0 \end{array}\right) $$

Prove that the intersection of any two ideals of \(A\) is an ideal of \(A\).

If \(n\) is a multiple of \(m\), then \(\mathbb{Z}_{m}\) is a homomorphic image of \(\mathbb{Z}_{n}\).

For any \(a \in A,\\{x \in A: a x=0\\}\) is an ideal (called the annihilator of \(a\) ). Furthermore, \(\\{x \in A: a x=0\) for every \(a \in A\\}\) is an ideal (called the annihilating ideal of A). If \(A\) is a ring with unity, its annihilating ideal is equal to \(\\{0\\}\).

Show that the set of all \(2 \times 2\) matrices of the form $$ \left(\begin{array}{ll} 0 & 0 \\ 0 & x \end{array}\right) $$ is a subring of \(\mathscr{M}_{2}(\mathbb{R})\), then prove this subring is isomorphic to \(\mathbb{R}\) For any integer \(k\), let \(k \mathbb{Z}\) designate the subring of \(\mathbb{Z}\) which consists of all the multiples of \(k\).

Let \(A\) and \(B\) be rings, and \(f: A \rightarrow B\) a homomorphism. Prove each of the following. \(f\) is injective iff its kernel is equal to \(\\{0\\}\)

Prove that each of the following is a homomorphism. Then describe its kernel and its range. \(h: \mathbb{R} \times \mathbb{R} \rightarrow \mathscr{M}_{2}(\mathbb{R})\) given by $$ h(x, y)=\left(\begin{array}{ll} x & 0 \\ 0 & y \end{array}\right) $$

Prove that if \(J\) is an ideal of \(A\) and \(1 \in J\), then \(J=A\)

If \(n\) is odd, there is an injective homomorphism from \(\mathbb{Z}_{2}\) into \(\mathbb{Z}_{2 n}\).

Prove that \(\mathbb{Z} \nsupseteq 2 \mathbb{Z} ;\) then prove that \(2 \mathbb{Z} \nsupseteq 3 \mathbb{Z}\). Finally, explain why if \(k \neq l\), then \(k \mathbb{Z} \nsupseteq l \mathbb{Z}\)

Let \(A\) be the set \(\mathbb{R} \times \mathbb{R}\) with the usual addition and the following "multiplication": $$ (a, b) \odot(c, d)=(a c, b c) $$ Granting that \(A\) is a ring, let \(f: A \rightarrow \mathscr{M}_{2}(\mathbb{R})\) be given by $$ f(x, y)=\left(\begin{array}{ll} x & 0 \\ y & 0 \end{array}\right) $$

Every subring of a field is an integral domain.

Explain why a field \(F\) can have no nontrivial ideals (that is, no ideals except \(\\{0\\}\) and \(\mathrm{F}\) ).

If a subring \(B\) of a field \(F\) is closed with respect to multiplicative inverses, then \(B\) is a field. ( \(B\) is then called a subfield of \(F\).)

The subset of \(\mathscr{H}_{2}(\mathbb{R})\) consisting of all matrices of the form $$ \left(\begin{array}{ll} 0 & 0 \\ 0 & x \end{array}\right) $$ is a subring of \(\mathscr{H}_{2}(\mathbb{R})\)

List all the homomorphisms from \(\mathbb{Z}_{2}\) to \(\mathbb{Z}_{4} ;\) from \(\mathbb{Z}_{3}\) to \(\mathbb{Z}_{6}\).

Let \(A\) be a ring, \(f: A \rightarrow A\) a homomorphism, and \(B=\\{x \in A: f(x)=x\\} .\) Then \(B\) is a subring of \(A\).

Give an example of a subring of \(P_{3}\) which is not an ideal.

The center of a ring \(A\) is the set of all the elements \(a \in A\) such that \(a x=x a\) for every \(x \in A\). Prove that the center of \(A\) is a subring of \(A\).

Give an example of a subring of \(\mathbb{Z}_{3} \times \mathbb{Z}_{3}\) which is not an ideal.