Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121220200601Counting subrings of $mathbb{Z}^n$ of non-zero co-rank16317211886810.30504/jims.2020.238412.1020ENS.ChimniDepartment of Mathematics, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St (M/C 249), Chicago, IL 60607.G.ChintaDepartment of Mathematics, The City College of New York, New York, NY 10031.R.Takloo-BighashDepartment of Mathematics, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St (M/C 249), Chicago, IL 60607.0000-0002-5340-2412Journal Article20200707In this paper we study subrings of $mathbb{Z^{n+k}}$ of co-rank $k$. We relate the number of such subrings $R$ with torsion subgroup $(mathbb{Z^{n+k}}/R)_{rm{tor}}$ of size $r$ to the number of full rank subrings of $mathbb{Z^n}$ of index $r$.http://jims.ims.ir/article_118868_57f0145a78e287eccee85f1c2bbf2a9d.pdf