Two matrices $D$ and $E$ over a field have the same nullspace if only if they are row-equivalent. Is the same true if those matrices are over the ring of integers ($\mathbb{Z}$) or integers mod a fixed positive integer ($\mathbb{Z}/(n \mathbb{Z})$)?
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$\begingroup$ The elementary row operation of multiplication by a constant must be by a unit in $\mathbb{Z}_n$ otherwise the operation won't be invertible. I have no idea if defining row equivalence with this constraint to row operations would still be equivalent to defining row equivalence as just having the same row-space. $\endgroup$– JoséCommented Mar 28 at 0:46
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$\begingroup$ $\mathbb{Z}_p$ usually refers to $p$-adic numbers. Please be explicit on your notation. $\endgroup$– YCorCommented Mar 28 at 9:58
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2$\begingroup$ Over $\mathbb{Z}$, having the same kernel is weaker. For instance, take one integer matrix, and multiply it by any number that is not $0$ or $1$. The kernel remains the same, but the two matrices are no longer row equivalent. $\endgroup$– Igor KhavkineCommented Mar 28 at 20:55
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1$\begingroup$ Row operations over $\mathbb Z$ give you the Hermite normal form. $\endgroup$– Emil JeřábekCommented Mar 29 at 6:39
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1$\begingroup$ For completeness, the analog of the Hermite normal form over the integers is the Howell normal form over $\mathbb{Z}/n\mathbb{Z}$. $\endgroup$– Igor KhavkineCommented Mar 29 at 9:03
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