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Let $R:=\mathbb{Z}[X_1,X_2,\dots,X_{mn}]$. Suppose $A=(f_{ij})$ is a $m\times n$ matrix with entries in $R$ such that

(1)there is no zero column in $A$;

(2)for each $i,j$, either $f_{ij}=0$ or $f_{ij}=X_k$ for some $k\in \{1,2,…,mn\}$;

(3)if $f_{ij}\neq 0$ and $f_{i'j'} \neq 0$, then $f_{ij} \neq f_{i'j'}$.

Is it true that if there exists nonzero $f_1,f_2,\dots,f_m$(here $f_i\neq 0$ for every $i$) in $R$ such that $(f_1\ f_2\ \dots\ f_m)A=0$, then we must have $m>n$?

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    @YACP, yes, posted 16 Oct, closed 17 Oct, deleted 19 Oct. On MO, people are expected to clean up their own questions, which OP made no effort to do. The customs at m.se are different.2013-02-12

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