When chracterizing the definition of unique factorization domain ring, the Hungerford's text, for example, states that UFD1 any nonzero nonunit element x is written as x=c_1. . .c_n. Does this mean any nonzero nonunit element is always written as a product of finitely many irreducible elements? I think it is not the case. Because if it were then it implies that any R has finitely many irreducible elements. So is it just for convenience's sake? So even if a nonzero nonunit element is a product of infinitely many irreducible elements, we can just put it as x=c_1. . .c_n? Even if it is uncountably infinitely many?