Exercises 1.10 Exercises
2.
Show that \(R\) is an integral domain if and only if the polynomial ring over \(R\text{,}\) \(R[X]\) is an integral domain.3.
Let \(F\) be a field. The set of polynomials in \(n\) variables over \(F\) is
\begin{align*}
F[X_1,X_2,\ldots,X_n]\amp=\bigg\{\sum a_{i_1\cdots i_r}X^{i_1}X^{i_2}\cdots X^{i_r}:i_j\in\N\cup\{0\},\\
\amp\quad\qquad\qquad a_{i_1\cdots i_r}\in F\text{ and all but finitely many }a_{i_1\cdots i_r}= 0\bigg\}
\end{align*}
The set \(F[X_1,\ldots,X_n]\) is defined inductively. When \(n=1\) it is a polynomial ring. Suppose we have defined \(R=F[X_1,\ldots,X_{n-1}]\text{.}\) Then \(F[X_1,\ldots,X_n]=R[X_n]\text{.}\) Show that it is a ring, in fact, an integral domain. Further show that \(F[X_1,\ldots,X_n]\) is an \(F\)-algebra.
4.
Let \(R\) be a ring and \(I\) be a right (resp., left) ideal of \(R\text{.}\) Show that \(\left(I^0\right)^0=I\) (refer to Example 1.6.10).5.
Let \(V\) be a finite-dimensional vector space over a field \(F\) and \(U\) be a subspace of \(V\text{.}\) Prove the following.
\begin{equation*}
\Hom_F(V,U)^0=\Hom_F(V/U,V)\quad\text{and}\quad\Hom_F(V/U,V)^0=\Hom_F(V,U)
\end{equation*}
6.
Show that there are no proper nontrivial two-sided ideals in the ring of \(n\times n\) matrices over a field \(F\text{.}\)7.
Let \(R\) be a ring. Show that if \(0\neq r\in R\) has no left inverse but has a right inverse then \(r\) has infinitely many right inverses.We show that if \(n>0\) then \(r\) has at least \(n+1\) right inverses.
- Assume that there exists \(n>0\) such that \(s_k\) is a right inverse of \(r\) for \(0\leq k\leq n\text{,}\) and \(s_k\neq s_\ell\) for \(k\neq\ell\text{.}\)
- For each \(k\in\{0,1,\ldots,n-1\}\) we put\begin{equation*} t_k=s_0+1-s_kr. \end{equation*}Show that \(rt_k=1\text{,}\) i.e., each \(t_k\) is a right inverse of \(r\text{.}\)
- Show that \(t_k\neq t_\ell\) for \(k\neq\ell\) and \(k,\ell\in\{0,1,\ldots,n-1\}\text{.}\)
Give an example of a ring and a nonzero element with no left inverse but infinitely many right inverses.
