I have been working on this problem where we consider the ring R={$T:P(\mathbb{R}$)$\rightarrow P(\mathbb{R})|T$ is a linear transformation}, where $P(\mathbb{R})$ is the set of polynomials with real coefficients and we let:
- $T_{0}:P(\mathbb{R})\rightarrow P(R)$ be the zero map where $T_{0}(p)=0$
- $D:P(\mathbb{R})\rightarrow P(R)$ be the differentiation map where $D(p)=p'$
- $J:P(\mathbb{R})\rightarrow P(R)$ be the integration map where $J(p)=\int_{0}^{x} p(t)dt$
- $E:P(\mathbb{R})\rightarrow P(R)$ be the evaluation map where $E(p)=p(0)$
- $I:P(\mathbb{R})\rightarrow P(R)$ be the identity map where $I(p)=p$
The first part of the question asked to show that $DJ=I$ thereby showing that $D$ has a right inverse and $I$ has a left inverse. This first part easily follows from the fundamental theorem of calculus and therefore it is true.
The second part asked to show that $EJ=T_{0}$ and $DE=T_{0}$ which is also true since, for some $p\in P(\mathbb{R})$, we have that $E(J(p))=E(P(x)-P(0))=P(0)-P(0)=0$ (here P denotes the anti-derivative of p), and $D(E(p))=D(p(0))=0$. This shows that $J$ is a right zero divisor while $D$ is a left zero divisor.
The final portion is where I have gotten stuck on, and it asks to show, using the first two parts, that $D$ does not have a left inverse nor is it a right zero divisor. It also asks to show that $J$ does not have a right inverse nor is it a left zero divisor.
So, far I have noticed that since a zero divisor (left or right) cannot be a unit implies that $D$ and $J$ cannot have left and right inverses respectively (here I have assumed that being a unit means having both a left and right inverse so please correct me if I am wrong). As for why $D$ cannot be a zero right divisor and $J$ cannot be a zero left divisor, I am having some trouble figuring this out so any hints or help would be appreciated.
Nothing with a right inverse can be a right zero divisor. For instance, let $S$ be such that $SD=T_0$. Then $S=SI=S(DJ)=(SD)J=T_0J=T_0$, showing that $S$ is $0$. This is why $D$ cannot be a right zero divisor. Similar reasoning holds for why $J$ cannot be a left zero divisor.