Exercise 11, § 2

Let A be a commutative ring and S a commutative monoid with identity element $e$. Suppose that an external law $(s, x) \mapsto x^s$ is given on A with S as domain of operators, such that, for all $s \in S$, the mapping $x \mapsto x^s$ is an automorphism of the ring A and $(x^s)^t = x^{st}$ for all $x \in A$ and $s$ and $t$ in S (which implies that $e$ is the identity operator for the external law considered). Then a multiplicative internal law of composition is defined on the A-module $A^{(S)}$, whose canonical basis is denoted by $(b_s)$, by the relation

where the $a_{s,t}$ belong to A.

Show that this law and addition on $A^{(S)}$ define a ring structure on this set, provided the $a_{s,t}$ satisfy the conditions

for all $s, t, u$ in S. The element $b_e$ is unit element of this ring if also

for all $s \in S$; in that case A can be identified with a subring of $A^{(S)}$ and if C denotes the subring of A consisting of the elements invariant under all the automorphisms $x \mapsto x^s$, $A^{(S)}$ is a C-algebra, called the crossed product of the ring A and the monoid S, relative to the factor system $(a_{s,t})$. If the factor system $(a_{s,t})$ is replaced by $\left( \frac{c_s c_t^s}{c_{st}} a_{s,t} \right)$, where, for all $s \in S$, $c_s$ is an invertible element of A and $c_e = 1$, the new crossed product obtained is isomorphic to the crossed product defined by the factor system $(a_{s,t})$.

If in particular A is taken to be a quadratic algebra over a ring C and S a cyclic group of order 2, say $\{e, s\}$, such that $x^s = \bar{x}$ for $x \in A$, show that every crossed product of A and S is a quaternion algebra over C.