Definition:Two-Sided Linear Combination in Ring

Definition
Let $R$ be a ring.

Let $(x_i)_{i\in I}$ be a family of elements of $R$.

A two-sided linear combination of the family is an element of the form:
 * $\displaystyle \sum_{i \mathop \in I} a_ix_ib_i$

where:
 * $(a_i)_{i\in I}$ and $(b_i)_{i\in I}$ are families in $R$ of finite support
 * $\sum$ denotes summation with finite support

Also see

 * Definition:Generated Ideal of Noncommutative Ring
 * Definition:Two-Sided Linear Combination in Bimodule