# Universal property of tensor product of $R$-algebras

Let $R$ be a commutative ring and $A_1,...,A_{n+1}$ be $R$-algebras.

Let $A_1\otimes_R\cdots\otimes_R A_{n+1}$ be equipped with the natural $R$-algebra structure.

Let $N$ be an $R$-algebra.

Let $\phi:A_1\times\cdots \times A_{n+1}\rightarrow N$ be a middle linear map such that $\phi(a_1b_1,...,a_{n+1}b_{n+1})=\phi(a_1,...,a_{n+1})\phi(b_1,...,b_{n+1})$

Then, there exists a unique $(R,R)$-bimodule & $R$-algebra homomorphism $\Phi:A_1\otimes_R ...\otimes_R A_{n+1}\rightarrow N$ such that $\Phi(a_1\otimes...\otimes a_{n+1})=\phi(a_1,...,a_{n+1})$.

Is there a name for maps such as $\phi$? $R$-multialgebra?

• @user26857 Another term is "balanced". That is multiadditive and $\phi(mr,n)=\phi(m,rn)$, but need not be $\phi(mr,n)=r\phi(m,n)$ – Rubertos Mar 19 '15 at 8:20
• Hungerford uses the term "middle linear" where as Dummit & Foote used both the terms "middle linear" and "$R$-balanced". For commutative algebras, the term "bilinear" is usually used. – Krish Mar 19 '15 at 12:52