Exercise 3, § 1

a) Give an example of a commutative field K and an isomorphism $\varphi$ from K to one of its subfield $K'$ such that K is finite-dimensional over $K'$. On the additive group $A = K\times K$, we define a ring structure by setting $(x, y)(x', y') = (xx', xy'+y\varphi (x'))$ (II, §1, p. 382, Exercise 7). The only left ideals of A are $\{(0,0)\},\{0\} \times K$, and A, but for every $K'$-linear subspace E of $K,\{0\} \times E$ is a right ideal of A. Deduce that A is left Artinian and left Noetherian but neither right Artinian nor right Noetherian. b) Give an example of a module of finite length whose endomorphism ring is neither left Artinian nor left Noetherian.

c) Using the same method as in a), give an example of a left and right Artinian ring whose left length is different from its right length.