Primitive Elements in Symmetric Algebras

Extract

Let-R be a commutative ring with 1, and let

be the symmetric algebra of an R-module M. We regard the isomorphisms S⁰(M) ≅ R and S¹(M) ≅ M a s identifications. There is a unique R-algebra homomorphism Δ : S(M) → S(M) ⊗_RS(M) (called the comultiplication) satisfying Δ(m) = m ⊗ 1 + 1 ⊗ m for all m ∊ M; any element x ∊ S(M) for which Δ(x) = x ⊗ 1 + 1 ⊗ x is said to be primitive. The set of all primitive elements in S(M) is denoted P(M).