Why is left ideal called right?

Question

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.

Answer 1

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.