Exercise 21, § 1

Let A be a ring, $\sigma$ an endomorphism of the ring A, and $d$ a nilpotent endomorphism of the additive group of A satisfying the relation $d(ab) =\sigma (a)d(b) +d(a)b$. a) Show that the set $S =\{X\}$ admits a calculus of left fractions (Exercise 18) on the ring $B = A[X]_{\sigma ,d}$ (VIII, p. 11). If $\sigma$ is injective, then the canonical homomorphism from B to $S^{-1}B$ is injective and we identify B with a subring of $S^{-1}B$. If $\sigma$ is bijective, then S also admits a calculus of right fractions on B, and every element of $S^{-1}B$ can be written uniquely as $\sum_{n\in\mathbf{Z}}a_nX^n$, where $(a_n)_{n\in\mathbf{Z}}$ is a family of elements of A with finite support. For $a\in A$, write the element $X^{-1}a$ of $S^{-1}B$ in this form.

b) Prove that the multiplication in the ring $A[X]_{\sigma ,d}$ extends by continuity to a multiplication on the set A[[X]] of formal power series in X with coefficients in A. The additive group A[[X]] endowed with this multiplication is a ring that we denote by $A[[X]]_{\sigma ,d}$ (or simply $A[[X]]_{\sigma}$ if $d= 0$). Give an example where a formal power series $u\in A[[X]]$ with constant term equal to 1 has neither a left nor a right inverse in the ring $A[[X]]_{\sigma ,d}$.

c) Show that the invertible elements of $A[[X]]_{\sigma}$ are those whose constant term is invertible in A.

d) Formulate and prove the analog of a) after replacing the ring $A[X]_{\sigma ,d}$ with the ring $A[[X]]_{\sigma ,d}$.

e) Suppose that A is a field and $\sigma$ an automorphism of A. Then $S^{-1}A[[X]]_{\sigma}$ (where $S =\{X\}$) is a field, and each of its elements can be written uniquely as $\sum_{n\in\mathbf{Z}}a_nX^n$, where $(a_n)_{n\in\mathbf{Z}}$ is a family of elements of A having only finitely many nonzero terms with index $<0$. We denote this field by $A((X))_{\sigma}$.

f ) Suppose that A is a commutative field. Describe the center of the field $A((X))_{\sigma}$. Deduce an example of a field of infinite degree over its center.