Algebras with a bilinear form, and Idempotent endomorphisms

Abstract

The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the canonical homomorphism $(k,1)\to(A,1_A)$ of unital $k$-algebras is a splitting monomorphism in the category of $k$-modules. Call the left inverses of this splitting monomorphism "weak augmentations" of the algebra. There is a category isomorphism between the category of $k$-algebras with a weak augmentation and the category of unital $k$-algebras $(A,b_A)$ with a bilinear form $b_A$ compatible with the multiplication of $A$, i.e., such that $b_A(x,y)=b_A(z,w)$ for all $x,y,z,w\in A$ for which $xy=zw$.