A short proof of Rédei's theorem (Q1206793)

Rédei's theorem states that the congruences on a finitely generated free commutative semigroup satisfy the ascending chain condition. The first short proof of this theorem was published by \textit{P. Freyd} [Proc. Am. Math. Soc. 19, 1003 (1968; Zbl 0191.01703)], it was based on the Hilbert Basis Theorem. The author's proof is based on \textit{L. E. Dickson}'s theorem [Am. J. Math. 35, 413-422 (1913; JFM 44.0220.02)]. The author mentions two other short proofs of Rédei's theorem.