Abstract algebra – Direct cancellation of products in Hopfian rings.

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 = $ where $ f_1: R times S to T $ Y $ f_2: R times S to S $. As $ S $ it's hopfian, therefore $ f_2 $ It is an isomorphic function. I wonder if the isomorphism of $ f $ does not imply isomorphism of $ f_1 $ Y $ f_2 $. Also, please let me know if this is the right approach or if I should see it in some other way.

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