Geodesic
On additions on the multiplicative monoid of integers
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 4, Tome 502 (2021), pp. 139-151

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We study modules over a certain generalized ring. This ring is a noncommutative tensor square of the ring of integers. The modules in the question are related to some interesting arithmetic problems. In particular they are related to solved Gauss' class number problem for imaginary quadratic fields. 
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     author = {A. L. Smirnov},
     title = {On additions on the multiplicative monoid of integers},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {139--151},
     publisher = {mathdoc},
     volume = {502},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_502_a5/}
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A. L. Smirnov. On additions on the multiplicative monoid of integers. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 4, Tome 502 (2021), pp. 139-151. http://geodesic.mathdoc.fr/item/ZNSL_2021_502_a5/