Counting subrings of $\mathbb{Z}^n$ of non-zero co-rank

Document Type: Original Article


1 Department of Mathematics, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St (M/C 249), Chicago, IL 60607.

2 Department of Mathematics, The City College of New York, New York, NY 10031.


In 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 $r$.


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