algebraprojection

An algebraprojection is a linear endomorphism P of an algebra A over a field F that satisfies two properties: idempotence P^2 = P and multiplicativity P(xy) = P(x) P(y) for all x, y in A. In this setting, the image E = P(A) is a subalgebra of A on which P acts as the identity.

Consequences of these conditions include that E is a subalgebra and the kernel K = ker P is a two-sided ideal. Every element a in A can be written uniquely as a = e + k with e in E and k in K, and P(a) = e. Thus A decomposes as a direct sum of subspaces A = E ⊕ K, with multiplication arranged so that K is an ideal and E is closed under multiplication.

Constructions and examples: If A splits as an algebra direct sum A ≅ E ⊕ F, the canonical projection onto E is an algebraprojection. Central idempotents offer another source: for a central idempotent e in A, the map P(x) = e x is an algebraprojection with image eA; when e is central, this map is multiplicative. Such projections provide a clean way to realize A as a combination of a preserved subalgebra and a complementary part.

Relation to broader concepts: algebraprojections generalize retractions in the category of algebras and relate to conditional expectations in operator algebras. They encode a structural decomposition of A into a subalgebra and an ideal and appear in areas such as representation theory, invariant substructure analysis, and module theory. The term is not universal in standard references, but the underlying idea is widely used under related notions.