
22 Algebraic Operads: An Algorithmic Companion
Let A be an associative algebra, and suppose that S ⊂ A. Recall that the
ideal of A generated by S, conventionally denoted by (S), is the smallest (by
inclusion) ideal of A that contains S as a subset. Explicitly, the ideal (S) is
the linear span of all elements r
1
sr
2
for all r
1
, r
2
∈ A, s ∈ S.
Definition 2.2.2.1 (Presentation by generators and relations). Suppose that
the algebra A is a quotient of the free algebra T (X) by some ideal I, and that
the ideal I is generated by the set S. In this case, we will say that the algebra
A is presented by generators X and relations S.
This way, working with finitely generated ...