A "close analogue" of (what I'm referring to as) Cohn's theorem on atomic factorizations in cancellative monoids (that is, Theorem 1 in the OP) is given by the unnumbered corollary on the bottom of p. 589 in P.M. Cohn's

• Torsion modules over free ideal rings, Proc. London Math. Soc. III. Ser. 17 (1967), 577-599.

After recalling (or introducing?) the notions of "atom" and "atomic ring" (the latter definition makes explicit reference to the monoid of regular elements, as can be seen from loc. cit., p. 587), Cohn proves that, if a ring $R$ is Morita-equivalent to a FIR, then the monoid of regular elements of $R$ is atomic: The proof relies on (i) Proposition 4.3 from the same paper (where Cohn shows that a FIR satisfies the ACCPR) and (ii) the observation that a FIR satisfies the ACCPR if and only if it satisfies the ACCPL.

The same idea (to combine the ACCPL and the ACCPR to prove atomicity) also appears in the proof of Theorem 2.8 from Cohn's Free ideal rings, which was published in the 1st issue of the 1st volume of J. Algebra (back in 1964). It seems that, at the time, Cohn had not yet coined the term "atom" and was rather using the term "prime" (with the same meaning), which was apparently common back then (cf. Cohn's 1963 TAMS paper on non-commutative unique factorization domains in Trans. AMS or R.E. Johnson's 1965 PAMS paper on unique factorization in principal right ideal domains).

It is perhaps worth noting that, in the 1964 paper, Cohn was not yet explicitly "thinking in monoids", to the contrary of the 1967 paper cited in the above; and none of these papers includes any reference to the work of other people building on the idea of combining the ACCPL and the ACCPL to prove atomicity.

A "close analogue" of (what I'm referring to as) Cohn's theorem on atomic factorizations in cancellative monoids (that is, Theorem 1 in the OP) is given by the unnumbered corollary on the bottom of p. 589 in P.M. Cohn's

• Torsion modules over free ideal rings, Proc. London Math. Soc. III. Ser. 17 (1967), 577-599.

After recalling (or introducing?) the notions of "atom" and "atomic ring" (the latter definition makes explicit reference to the monoid of regular elements, as can be seen from loc. cit., p. 587), Cohn proves that, if a ring $R$ is Morita-equivalent to a FIR, then the monoid of regular elements of $R$ is atomic: The proof relies on (i) Proposition 4.3 from the same paper (where Cohn shows that a FIR satisfies the ACCPR) and (ii) the observation that a FIR satisfies the ACCPR if and only if it satisfies the ACCPL.

The same idea (to combine the ACCPL and the ACCPR to prove atomicity) also appears in the proof of Theorem 2.8 from Cohn's Free ideal rings, which was published in the 1st issue of the 1st volume of J. Algebra (back in 1964). It seems that, at the time, Cohn had not yet coined the term "atom" and was rather using the term "prime" (with the same meaning), which was apparently common back then (cf. Cohn's 1963 TAMS paper on non-commutative unique factorization domains in Trans. AMS or R.E. Johnson's 1965 PAMS paper on unique factorization in principal right ideal domains).

It is perhaps worth noting that, in the 1964 paper (cited above), Cohn was not yet explicitly "thinking in monoids", to the contrary of the 1967 paper; and none of these papers includes any reference to the work of other people building on the idea that "ACCPL & ACCPR $\implies$ atomicity".

It turns out that the earliest appearance of a "close analogue" of (what I'm referring to as) Cohn's theorem on atomic factorizations in cancellative monoids (that is, Theorem 1 in the OP) may well have been Theorem 2.8 in Cohn's

• Torsion modules over free ideal rings, Proc. London Math. Soc. III. Ser. 17 (1967), 577-599.

After recalling (or introducing?) the notions of "atom" and "atomic ring" (the latter definition makes explicit reference to the monoid of regular elements, as can be seen from loc. cit., p. 587), Cohn proves that, if a ring $R$ is Morita-equivalent to a FIR, then the monoid of regular elements of $R$ is atomic: The proof relies (i) on Proposition 4.3 from the same paper, where Cohn shows that a FIR satisfies the ACCPR; and (ii) on the observation that, by symmetry, a FIR satisfies the ACCPR if and only if it satisfies the ACCPL.

The same idea (to combine the ACCPL and the ACCPR to prove atomicity) also appears in the proof of Theorem 2.8 from Cohn's Free ideal rings, which was published in the 1st issue of the 1st volume of J. Algebra (back in 1964). It seems that, at the time, Cohn had not yet coined the term "atom" and was rather using the term "prime" (with the same meaning), as he had already done in his 1963 TAMS paper on non-commutative unique factorization domains.

For the record: In the 1964 paper, Cohn was not yet explicitly "thinking in monoids", to the contrary of the 1967 paper cited in the above.

A "close analogue" of (what I'm referring to as) Cohn's theorem on atomic factorizations in cancellative monoids (that is, Theorem 1 in the OP) is given by the unnumbered corollary on the bottom of p. 589 in P.M. Cohn's

• Torsion modules over free ideal rings, Proc. London Math. Soc. III. Ser. 17 (1967), 577-599.

After recalling (or introducing?) the notions of "atom" and "atomic ring" (the latter definition makes explicit reference to the monoid of regular elements, as can be seen from loc. cit., p. 587), Cohn proves that, if a ring $R$ is Morita-equivalent to a FIR, then the monoid of regular elements of $R$ is atomic: The proof relies on (i) Proposition 4.3 from the same paper (where Cohn shows that a FIR satisfies the ACCPR) and (ii) the observation that a FIR satisfies the ACCPR if and only if it satisfies the ACCPL.

The same idea (to combine the ACCPL and the ACCPR to prove atomicity) also appears in the proof of Theorem 2.8 from Cohn's Free ideal rings, which was published in the 1st issue of the 1st volume of J. Algebra (back in 1964). It seems that, at the time, Cohn had not yet coined the term "atom" and was rather using the term "prime" (with the same meaning), which was apparently common back then (cf. Cohn's 1963 TAMS paper on non-commutative unique factorization domains in Trans. AMS or R.E. Johnson's 1965 PAMS paper on unique factorization in principal right ideal domains).

It is perhaps worth noting that, in the 1964 paper, Cohn was not yet explicitly "thinking in monoids", to the contrary of the 1967 paper cited in the above; and none of these papers includes any reference to the work of other people building on the idea of combining the ACCPL and the ACCPL to prove atomicity.