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edited Nov 3, 2014 at 22:45

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Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.

Consider a equivalent relation on $R\mathbb{Z}^N$ defined by $a\sim b$ if $a-b\in \mathbb{Z}^M$ for any $a,~b\in R\mathbb{Z}^N$. Denote the set of equivalent classes as $(R\mathbb{Z}^N)/\mathbb{Z}^M$. Similarly, we have the notion of $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$. Both $(R\mathbb{Z}^N)/\mathbb{Z}^M$ and $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$ form groups under addition.

QuestionQuestions:

(1) Is $(R\mathbb{Z}^N)/\mathbb{Z}^M$ isomorphic to $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$?

(2) If (1) is not true, Is the cardinality $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

(This is posted on both Math Overflow and Math Stack Exchange.)

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edited Nov 3, 2014 at 2:58

imaboy

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Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N+\mathbb{Z}^M^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.

LetConsider a equivalent relation on $(R\mathbb{Z}^N+\mathbb{Z}^M)=\{R\Lambda+\Lambda'|\Lambda\in\mathbb{Z}^N,~\Lambda'\in\mathbb{Z}^M\}$$R\mathbb{Z}^N$ defined by which forms a group under addition.

Note that$a\sim b$ if $(R\mathbb{Z}^N+\mathbb{Z}^M)$ has a subgroup$a-b\in \mathbb{Z}^M$ for any $\mathbb{Z}^M$, and hence we can construct$a,~b\in R\mathbb{Z}^N$. Denote the quotient groupset of equivalent classes as $(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M$$(R\mathbb{Z}^N)/\mathbb{Z}^M$. Similarly Similarly, we have the notion of $(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N$$(R^T\mathbb{Z}^M)/\mathbb{Z}^N$. Both $(R\mathbb{Z}^N)/\mathbb{Z}^M$ and $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$ form groups under addition.

Question:

(1) Is $(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M$$(R\mathbb{Z}^N)/\mathbb{Z}^M$ isomorphic to $(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N$$(R^T\mathbb{Z}^M)/\mathbb{Z}^N$?

(2) If (1) is not true, Is the cardinality $|(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N|$$|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.

Let $(R\mathbb{Z}^N+\mathbb{Z}^M)=\{R\Lambda+\Lambda'|\Lambda\in\mathbb{Z}^N,~\Lambda'\in\mathbb{Z}^M\}$ which forms a group under addition.

Note that $(R\mathbb{Z}^N+\mathbb{Z}^M)$ has a subgroup $\mathbb{Z}^M$, and hence we can construct the quotient group $(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M$. Similarly, we have the notion of $(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N$

Question:

(1) Is $(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M$ isomorphic to $(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N$?

(2) If (1) is not true, Is the cardinality $|(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.

Consider a equivalent relation on $R\mathbb{Z}^N$ defined by $a\sim b$ if $a-b\in \mathbb{Z}^M$ for any $a,~b\in R\mathbb{Z}^N$. Denote the set of equivalent classes as $(R\mathbb{Z}^N)/\mathbb{Z}^M$. Similarly, we have the notion of $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$. Both $(R\mathbb{Z}^N)/\mathbb{Z}^M$ and $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$ form groups under addition.

Question:

(1) Is $(R\mathbb{Z}^N)/\mathbb{Z}^M$ isomorphic to $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$?

(2) If (1) is not true, Is the cardinality $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

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asked Nov 3, 2014 at 2:51

imaboy

31
4

Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.

Let $(R\mathbb{Z}^N+\mathbb{Z}^M)=\{R\Lambda+\Lambda'|\Lambda\in\mathbb{Z}^N,~\Lambda'\in\mathbb{Z}^M\}$ which forms a group under addition.

Note that $(R\mathbb{Z}^N+\mathbb{Z}^M)$ has a subgroup $\mathbb{Z}^M$, and hence we can construct the quotient group $(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M$. Similarly, we have the notion of $(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N$

Question:

(1) Is $(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M$ isomorphic to $(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N$?

(2) If (1) is not true, Is the cardinality $|(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N|$?

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Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N+\mathbb{Z}^M^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N|$?