Exercise 1, § 3

Let p be a prime number, G the free group on two generators X and Y, $F_p[G]$ the group algebra of G over $F_p$. Put $Z = XY - YX$ and denote by $\alpha$ the two-sided ideal of $F_p[G]$ generated by the elements $ZX - XZ, ZY - YZ$ and $Z^2$. Putting $A = F_p[G]/\alpha$, denote by $\mathfrak{z}$ the two-sided ideal of A generated by the image of Z in A (the images of X, Y and Z in A will again be denoted by X, Y, Z). a) Let $a$ be an element of A. Prove that there exist two families of finite support of elements of $F_p$, say $(p_{\alpha,\beta})_{(\alpha,\beta) \in \mathbf{Z}^2}$ and $(q_{\alpha,\beta})_{(\alpha,\beta) \in \mathbf{Z}^2}$, uniquely determined by the condition

Deduce that the quotient algebra $A/\mathfrak{z}$ is commutative and an integral domain, and that the ideal $\mathfrak{z}$ is contained in the center of A. b) For every pair $(a, b)$ of elements of A, there exists an element c of A such that one has Deduce that the set $A^p$ of p-th powers of elements of A is contained in the center of A. c) The multiplicative subset $S = A - \mathfrak{z}$ of A admits a calculus of right and left fractions (II, § 2, Exercise 22). Denote by B the ring of fractions of A thus constructed. (To verify, for example, that for every $a$ in A and every $s$ in S, there exist $b \in A$ and $t \in S$ such that $at = sb$, take $b = s^{p-1} a, t = s^p$ and use b).) d) Prove that the canonical mapping of A into B is injective. (Observe that the kernel of this mapping is contained in $\mathfrak{z}$ and reason as in a.).

e) The center of B contains Z. If m is the two-sided ideal of B generated by Z, one has $m^2 = 0$. f) The ideal m is the set of non-invertible elements of B, and the field B/m is isomorphic to the field $F_p(X, Y)$. g) Prove that there does not exist a subfield of B whose image in B/m is all of B/m.

In Exercises 2 to 5, the (commutative) topological rings are supposed linearly topologized. If A and B are two topological rings and B is an A-algebra, one says that B is a topological A-algebra if the homomorphism from A into B which defines the algebra structure is continuous.