Abstract
A space of pseudoquotients, denoted by $\mathcal{B}(X,S)$, is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) \sim (y,g)$ when $gx=fy$. If $X$ is a ring and elements of $S$ are ring homomorphisms, then $\mathcal{B}(X,S)$ is a ring. We show that, under natural conditions, a $\hbar$ -Jordan centralizer on $X$ has a unique extension to a $\hbar$-Jordan centralizer on $\mathcal{B}(X,S)$
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