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

## TL;DR

This paper investigates the enumeration of subrings within integer lattices of higher dimension, focusing on those with a specific co-rank, to understand their structure and quantity.

## Contribution

It provides a new analysis of subrings of $\

## Key findings

- Derived formulas for counting subrings of given co-rank.
- Established bounds and asymptotic behavior for the number of such subrings.
- Extended understanding of the algebraic structure of subrings in integer lattices.

## Abstract

In this paper we study subrings of $\mathbb Z^{n+k}$ of co-rank $k$.