definition. tensor algebra [ca-001H]
definition. tensor algebra [ca-001H]
Let \(A\) be a free \(R\)-algebra generated by module \(M\), let \(\iota : M \to A\) denote the map from \(M\) to \(A\). An tensor algebra over \(M\) (or "of \(M\)") \(T\) is the ring quotient of the free \(R\)-algebra generated by \(M\), by the equivalence relation satisfying:
- for all \(a, b\) in \(M\), \(\iota (a + b) \sim \iota (a) + \iota (b)\).
- for all \(r\) in \(R\), \(a\) in \(M\), \(\iota (r \bullet a) \sim r * \iota (a)\).