A ring $ R $ is called to be & # 39; Hopfian & # 39; if each ring homomorphism of $ R $ on $ R $ it is an automorphism of $ R $

Question:

Dice $ R $, $ S $, $ T $, Hopfian rings and

$$ R times S cong T times S $$

It implies that there is an isomorphism.

$$ R cong T $$

My approach:

Leave $ X1, X2, Y in C $.

According to the product rule in the Category theory for each object Y and pair of morphisms

$$ f1: Y to X1, space space f2: Y → X2 $$

there is a unique map $$ f: Y → X1 × X2 $$ where $ f =

Dice $ f: R times S to T times S $, Thus $ f =

P.S: This is my first question in the math exchange, help me correct any mistakes you may have made here.