91Ó°ÊÓ

Let \(R\) be a ring such that \(x^{2}=x\) for all \(x \in R\). Show that \(R\) is commutative.

Prove in detail that the units of a ring form a multiplicative group.

Let \(R\) be a commutative ring. An ideal \(P\) is said to be a prime ideal if \(P \neq R\), and whenever \(a, b \in R\) and \(a b \in P\) then \(a \in P\) or \(b \in P\). Show that a non-zero ideal of \(Z\) is prime if and only if it is gencrated by a prime number.

Let \(R\) be a commutative ring. A map D: \(R \rightarrow R\) is called a derivation if \(D(x+y)=D x+D y\), and \(D(x y)=(D x) y+x(D y)\) for all \(x, y \in R .\) If \(D_{1}, D_{2}\) are derivations, define the bracket product $$ \left[D_{1}, D_{2}\right]=D_{1}=D_{2}-D_{2}=D_{1} . $$ Show that \(\left[D_{1}, D_{2}\right]\) is a derivation.