Math 361, Spring 2021, Assignment 3

Carefully define the following terms, then give one example and one non-example of each:[edit]

Laws of sign.
Theorem concerning $\phi(0_R)$ (when $\phi:R\rightarrow S$ is a ring homomorphism).
Example to show that the instinctive conjecture regarding $\phi(1_R)$ need not be true.
Theorem relating kernels to ideals.

Section 18, problems 14, 15, 16, 17, 18, 19, 20, and 22.
Suppose that $\phi:R\rightarrow S$ is a ring homomorphism, and that $T$ is a subring of $R$. Prove that $\phi[T]$ is a subring of $S$. Show also that if $R,S,T$ and $\phi$ are all unital, then so is $\phi[T]$.
Suppose that $\phi:R\rightarrow S$ is a ring homomorphism, and that $U$ is a subring of $S$. Prove that $\phi^{-1}[U]$ is a subring of $R$. Show also that if $R,S,U,$ and $\phi$ are all unital, then so is $\phi^{-1}[U]$.
Let $R$ be the matrix ring $M_2(\mathbb{R})$, and let $m=\begin{bmatrix}1&0\\0&0\end{bmatrix}$. We can describe the left principal ideal $Rm$ as follows: an arbitrary left multiple of $m$ has the form $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}1&0\\0&0\end{bmatrix}=\begin{bmatrix}a&0\\c&0\end{bmatrix}$, so $Rm=\left\{\begin{bmatrix}a&0\\c&0\end{bmatrix}\,\middle|\,a,b\in\mathbb{R}\right\}.$ Make a similar description of the right principal ideal $mR$. In this example, is $Rm=mR$? Is either of these sets a two-sided ideal?
With notation as in the previous problem, can you find any matrix $m$ for which $Rm$ will be a two-sided ideal?
Take $R=\mathbb{R}$, the real number system with its usual operations. Concretely describe the principal ideal $\left\langle x\right\rangle$ for any real number $x$. (Hint: you will always obtain one of only two possible sets.)

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