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.