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The category of all $k$-algebras with a bilinear form, whose objects are allpairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is abilinear mapping, is equivalent to the category of unital $k$-algebras $A$ forwhich the canonical homomorphism $(k,1)\to(A,1_A)$ of unital $k$-algebras is asplitting monomorphism in the category of $k$-modules. Call the left inversesof this splitting monomorphism "weak augmentations" of the algebra. There is acategory isomorphism between the category of $k$-algebras with a weakaugmentation and the category of unital $k$-algebras $(A,b_A)$ with a bilinearform $b_A$ 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 $xy=zw$.

Algebras with a bilinear form, and Idempotent endomorphisms