# About bimodules $k$-algebras

I am studying representations from the book of Assem. There are the next concepts:

My question: for the author, $A$ is a ring with $k$-space vector structure, so $A$ has $2$ binary operations. Also $M$ is other $k$-space vector with one binary operation by definition of module, and it is not possible to establish a "multiplication" $m_1*m_2$; then how must it interpret $A_i=M_{ii}$?

• $A_i$ is naturally an $(A_i, A_i)$-bimodule via left and right multiplication. The equality is intended to mean as bimodules. Commented Jul 13, 2017 at 21:33
• The definition of bimodule as given is incomplete: it is important that the left and right actions of K on M obtained from those of A and B, respectively, coincide. Commented Jul 14, 2017 at 2:05
• By the way, Ibrahim Assem has written at least two books: please, when referring to a book, provide a complete bibliographical reference. Commented Jul 14, 2017 at 4:38

The ring $A_i$ is itself an $A_i$-$A_i$-bimodule, by letting both multiplications be the ring multiplication of $A_i$. So the author is assuming that the bimodule $M_{ii}$ is $A_i$ with this bimodule structure for each $i$.