Why is left ideal called right? For operations between elements of an algebraic structure:


*

*If $a \cdot b = c$, then $a$ is a left divisor of $c$;

*If $a \cdot b = 0$, then $a$ is a left zero divisor;

*If $a \cdot b = e$, then $a$ is a left inverse of $b$;

*...


For operations between an element and an algebraic structure S:


*

*The map $a \cdot S$ is a left translation of an element $a$ (N.Bourbaki);

*If $a \cdot S$ is the identity permutation of $S$, then $a$ is a left identity;

*If $a \cdot S$ is an injection, then $a$ is left cancellable;

*...


For operation between algebraic structures:


*

*In $A \oplus S$: $A$ is a left summand;

*In $A \times S$: $A$ is a left operand;

*...


Why is a left ideal (in the sense above) $a \cdot S$ ($A \cdot S$ = $A$) called a right ideal of $S$?
This never bothered me until I started finding connections between the terms.
For example, if left associates are elements that generate the same left ideal, then $a$ and $b$ are left associates if and only if $a$ and $b$ are right divisors of each other in a ring with unity.
This sounds like there is some mechanic that switches operands, but it is merely a confusing naming convention.

One defining property of a right ideal $A\subseteq S$ is that $As\subseteq A$ for any $s\in S$. In words, $A$ is closed under right multiplication.
So it makes sense to call it a right ideal. That is not to say it is senseless to call it a left ideal. Some times when establishing a convention one has to choose between two (or more) options, and it's not always possible to make a perfect choice.
