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binary operation on R, (a/b)/c = (a/c) / (b/c) and (a/b)*c = (a*c) / (b*c)

Source: 2009 Belarus TST 8.1

November 8, 2020
algebraBinary operationOperation

Problem Statement

On R a binary algebraic operation ''*'' is defined which satisfies the following two conditions: i) for all a,b∈Ra,b \in R, there exists a unique x∈Rx \in R such that x∗a=bx *a=b (write x=b/ax=b/a) ii) (a∗b)∗c=(a∗c)∗(b∗c)(a*b)*c= (a*c)* (b*c) for all a,b,c∈Ra,b,c \in R a) Is this operation necesarily commutative (i.e. a∗b=b∗aa*b=b*a for all a,b∈Ra,b \in R) ? b) Prove that (a/b)/c=(a/c)/(b/c)(a/b)/c = (a/c) / (b/c) and (a/b)∗c=(a∗c)/(b∗c)(a/b)*c = (a*c) / (b*c) for all a,b,c∈Ra,b,c \in R.
A. Mirotin
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