# Article

 Title: Bases of minimal elements of some partially ordered free abelian groups  (English) Author: Příhoda, Pavel Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 44 Issue: 4 Year: 2003 Pages: 623-628 . Category: math . Summary: In the present paper, we will show that the set of minimal elements of a full affine semigroup $A\hookrightarrow \Bbb N^k_0$ contains a free basis of the group generated by $A$ in $\Bbb Z^k$. This will be applied to the study of the group $\text{\rm K}_0(R)$ for a semilocal ring $R$. Keyword: full affine semigroups Keyword: partially ordered abelian groups Keyword: semilocal rings Keyword: direct sum decompositions MSC: 06F20 MSC: 16D40 MSC: 16D70 MSC: 16E20 MSC: 20F60 MSC: 20M14 idZBL: Zbl 1101.16010 idMR: MR2062878 . Date available: 2009-01-08T19:31:38Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119416 . Reference: [1] Bruns W., Herzog J.: Cohen-Macaulay rings.Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993. Zbl 0909.13005, MR 1251956 Reference: [2] Facchini A.: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules.Progress in Mathematics 197, Birkhäuser, 1998. Zbl 0930.16001, MR 1634015 Reference: [3] Facchini A., Herbera D.: $K_0$ of a semilocal ring.J. Algebra 225 1 (2000), 47-69. Zbl 0955.13006, MR 1743650 Reference: [4] Facchini A., Herbera D.: Projective modules over semilocal rings.in: D.V. Huynh (ed.) et al., Algebra and its Applications: Proceedings of the International Conference, Contemp. Math. 259, 2000, 181-198. Zbl 0981.16003, MR 1778501 Reference: [5] Goodearl K.R.: Partially ordered abelian groups with interpolation.Mathematical Surveys and Monographs no. 20, Amer. Math. Soc., 1986. Zbl 0589.06008, MR 0845783 .

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