# What's the English name for this algebraic operation? $ab=ac → b=c$ [duplicate]

In Hebrew, the operation of deducing $$ab=ac \Rightarrow b=c$$ is called "צימצום" [tsimtsum] which translates roughly to "elimination" or "reduction"

More specifically for non-abelian groups the above formula would be "Right-elimination" or "Right-sided elimination", and $$ba=ca \Rightarrow b=c$$ would be "Left elimination"

What is the accepted terminology for this in English?

– user376921
Commented Dec 20, 2019 at 10:06
• Just saying: This deduction is possible only if you know for a fact that $a ≠ 0$ $$ba=bc$$ $$a(b-c)=0$$ $$a = 0 \ or \ b=c$$ And your question has nothing to do with "Abstract Algebra". Please remove that Tag. Commented Dec 20, 2019 at 10:09
• Are you sure your terminology has correct chiralities? You've said we call elimination of a factor from the left right-elimination.
– J.G.
Commented Dec 20, 2019 at 10:24
• @RishiNandhaVanchi I'm currently studying a course called "Algebraic Structures" which is the topic of this question, but I couldn't find an "Algebraic Structures" tag, so I thought abstract-algebra would be a close enough fit. is this not the case? Commented Dec 20, 2019 at 10:54
• The 'algebra' tag is not allowed. Please choose a more specific tag, for instance abstract-algebra or algebra-precalculus. Commented Dec 20, 2019 at 11:02

If you're talking about groups, you can describe the operation as pre-multiplying (or post-multiplying) by $$a^{-1}$$. In monoids or semi-groups with the cancellative property, you cannot describe it like that because $$a^{-1}$$ does not formally exist, so you just call it applying the cancellative property.