It seems natural to try to generalize the group ring construction to algebraic structures other than groups and rings. We do so here.
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Let A and B be general algebras (in the sense of universal algebra.) What would the construction of A[B] be?
Well, we let B’s elements form a basis; specifically, we look at formal linear combinations of elements of B with scalars in A. We can “include” A and B in these by adjoining identities to A and B, which we can do without penalty (if a magma M has an identity already then we’re good, otherwise we can define an object e outside M such that ee = e and em = me = m for all m in M, and “rename” M to the union of M and {e}.) Actually, though, would the identities thing be an issue with multiple operations?
If we say that we don’t care about identities and/or embedding copies of A and B in our construction, then let’s just take A[B] to be the set of formal linear combinations. What operations can we define on this / what structure can we impose?
Well, look at the standard operations on a group ring for comparison. We define addition on formal linear combinations based purely off of addition on A, so in general we can do that for any operation o on A. But then, if A supports multiplication (as in the case of a group ring), this doesn’t agree with the standard definition of multiplication of linear combinations. So we should label specific operation symbols on A and B to distinguish different definitions for how those operations apply to the linear combinations. To make this as direct as possible a generalization of the standard construction, we assume based on this discussion that A has an addition and multiplication. Then, we can define addition on A[B] based off of addition on A. What about multiplication? Well, that requires multiplication on A as well as multiplication on B. So we get back anyway to the standard signatures and the standard construction for them.
So the group ring is in fact the most “natural” generalized construction of this sort — or rather, the most “natural” generalization would be to change the axioms of the algebraic structures A and B. So we could for example consider R[M] with M a monoid and R a near-ring, or something like that.
What about if we want to include as a valid operation taking the negative of an element of A[B]? This just requires additive inverses of elements of A. What if we want to include A and B “naturally” as subsets of A[B] (and let’s leave aside for now the question of adjoining identity elements)? Then, we’d assume that A has a multiplicative identity to have B embeddable (with every element b of B corresponding to 1b in A[B]), and we’d assume that B has an identity too to have A embeddable (with every element a of A corresponding to ae in A[B].) Without identities, we can also just pick elements of A and B (assuming both sets are nonempty) and declare the corresponding bijections to be canonical for embedding A and B, although then the algebra would be different. If we want things “closer” to the standard construction, the most “natural” generalization is to have A[B] with A and B the same signatures as for a group ring, where:
1. A has additive inverses
2. A has multiplicative identity
3. B has identity
Adjoining identities to embed copies in the presence of multiple operations is actually probably a more general topic that deserves its own future post.
