000148932 001__ 148932 000148932 005__ 20250123152145.0 000148932 0247_ $$2doi$$a10.1007/s00029-024-01002-9 000148932 0248_ $$2sideral$$a142122 000148932 037__ $$aART-2025-142122 000148932 041__ $$aeng 000148932 100__ $$aAntoine, Ramon 000148932 245__ $$aThe Cuntz semigroup of a ring 000148932 260__ $$c2025 000148932 5060_ $$aAccess copy available to the general public$$fUnrestricted 000148932 5203_ $$aFor any ring R, we introduce an invariant in the form of a partially ordered abelian semigroup built from an equivalence relation on the class of countably generated projective modules. We call the Cuntz semigroup of the ring R. This construction is akin to the manufacture of the Cuntz semigroup of a C*-algebra using countably generated Hilbert modules. To circumvent the lack of a topology in a general ring R, we deepen our understanding of countably projective modules over R, thus uncovering new features in their direct limit decompositions, which in turn yields two equivalent descriptions of . The Cuntz semigroup of R is part of a new invariant which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both and in a number of interesting situations, such as unit-regular rings, semilocal rings, and in the context of nearly simple domains. We also relate our construcion to the Cuntz semigroup of a C*-algebra. 000148932 540__ $$9info:eu-repo/semantics/openAccess$$aby$$uhttp://creativecommons.org/licenses/by/3.0/es/ 000148932 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/publishedVersion 000148932 700__ $$aAra, Pere 000148932 700__ $$0(orcid)0000-0001-9442-1583$$aBosa, Joan$$uUniversidad de Zaragoza 000148932 700__ $$aPerera, Francesc 000148932 700__ $$aVilalta, Eduard 000148932 7102_ $$12006$$2005$$aUniversidad de Zaragoza$$bDpto. Matemáticas$$cÁrea Algebra 000148932 773__ $$g31, 1 (2025), [46 pp.]$$pSel. math., New ser.$$tSelecta Mathematica-New Series$$x1022-1824 000148932 8564_ $$s856839$$uhttps://zaguan.unizar.es/record/148932/files/texto_completo.pdf$$yVersión publicada 000148932 8564_ $$s1326291$$uhttps://zaguan.unizar.es/record/148932/files/texto_completo.jpg?subformat=icon$$xicon$$yVersión publicada 000148932 909CO $$ooai:zaguan.unizar.es:148932$$particulos$$pdriver 000148932 951__ $$a2025-01-23-14:47:19 000148932 980__ $$aARTICLE