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Bijective 1-cocycles, braces, and non-commutative prime factorization. (English) Zbl 1508.16043

The algebraic structure of the brace, a generalisation of the classical Jacobson radical rings, has been introduced by the author in [J. Algebra 307, No. 1, 153–170 (2007; Zbl 1115.16022)]. One of the motivations for studying braces is their interplay with non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation, a fundamental equation of mathematical physics (see [V. G. Drinfel’d, Lect. Notes Math. 1510, 1–8 (1992; Zbl 0765.17014)]).
In the paper under review, the author focuses on quasirings, namely additive abelian groups \(\left(A,+\right)\) endowed with a multiplication (written as juxtaposition) satisfying the following identities \begin{align*} 0a &=0\\ a\left(b + c\right) &= ab + ac\\ \left(ab + a + b\right)c &= a\left(bc\right) + ac + bc, \end{align*} for all \(a,b,c\in A\). By defining the operation \(\circ\) given by \(a\circ b:= ab + a + b\), one has that \(\left(A, \circ \right)\) is a monoid; if \(\left(A, \circ \right)\) is a group, then \(A\) is a brace. Consistently with braces, quasirings are equivalent to bijective \(1\)-cocycles \(M\to A\) from a monoid \(M\) onto an \(M\)-module \(A\).
The author shows that a specific class of lattice-ordered quasirings characterises the divisor groups of non-commutative smooth algebraic curves. Moreover, the adjoint monoid structure extends the multiplication of fractional ideals of a hereditary noetherian ring to the set of all divisors. Besides, he provides a description of the multiplication of divisors as an extension of the functional representation of fractional ideals given in [the author and Y. Yang, J. Algebra 468, 214–252 (2016; Zbl 1400.20058)].

MSC:

16T25 Yang-Baxter equations
14A22 Noncommutative algebraic geometry
06F05 Ordered semigroups and monoids
20M30 Representation of semigroups; actions of semigroups on sets
11M55 Relations with noncommutative geometry
16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
20F36 Braid groups; Artin groups
05E18 Group actions on combinatorial structures
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