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Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I mean an element $e\ne 0$ such that for every positive integer $n$ there is an element f such that $e=nf$.

As Darji points out, another way to ask the question is this: Suppose $e\in R$ has the property that for all positive integers $n$, $e$ is congruent to an integer mod $nR$. Must $e$ be an integer?

Note: I previously posted this to Math StackExchange here: http://math.stackexchange.com/questions/71031/a-question-about-the-additive-group-of-a-finitely-generated-integral-domain

TO SUMMARIZE: Qing Liu showed that in fact any non-integer rational in $R$ determines a divisible element of $(R,+)/(\mathbb{Z},+)$, and Wilberd van der Kallen showed that all divisible elements arise in this way. I wish I could accept both answers.

Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I mean an element $e\ne 0$ such that for every positive integer $n$ there is an element f such that $e=nf$.

As Darji points out, another way to ask the question is this: Suppose $e\in R$ has the property that for all positive integers $n$, $e$ is congruent to an integer mod $nR$. Must $e$ be an integer?

Note: I previously posted this to Math StackExchange here: https://math.stackexchange.com/questions/71031/a-question-about-the-additive-group-of-a-finitely-generated-integral-domain

TO SUMMARIZE: Qing Liu showed that in fact any non-integer rational in $R$ determines a divisible element of $(R,+)/(\mathbb{Z},+)$, and Wilberd van der Kallen showed that all divisible elements arise in this way. I wish I could accept both answers.

Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I mean an element $e\ne 0$ such that for every positive integer $n$ there is an element f such that $e=nf$.

As Darji points out, another way to ask the question is this: Suppose $e\in R$ has the property that for all positive integers $n$, $e$ is congruent to an integer mod $nR$. Must $e$ be an integer?

Note: I previously posted this to Math StackExchange here: http://math.stackexchange.com/questions/71031/a-question-about-the-additive-group-of-a-finitely-generated-integral-domain

TO SUMMARIZE: Qing Liu showed that in fact any rational in $R$ determines a divisible element of $(R,+)/(\mathbb{Z},+)$, and Wilberd van der Kallen showed that all divisible elements arise in this way. I wish I could accept both answers.

Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I mean an element $e\ne 0$ such that for every positive integer $n$ there is an element f such that $e=nf$.

As Darji points out, another way to ask the question is this: Suppose $e\in R$ has the property that for all positive integers $n$, $e$ is congruent to an integer mod $nR$. Must $e$ be an integer?

Note: I previously posted this to Math StackExchange here: http://math.stackexchange.com/questions/71031/a-question-about-the-additive-group-of-a-finitely-generated-integral-domain

TO SUMMARIZE: Qing Liu showed that in fact any non-integer rational in $R$ determines a divisible element of $(R,+)/(\mathbb{Z},+)$, and Wilberd van der Kallen showed that all divisible elements arise in this way. I wish I could accept both answers.

Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I mean an element $e\ne 0$ such that for every positive integer $n$ there is an element f such that $e=nf$.

As Darji points out, another way to ask the question is this: Suppose $e\in R$ has the property that for all positive integers $n$, $e$ is congruent to an integer mod $nR$. Must $e$ be an integer?

Note: I previously posted this to Math StackExchange here: http://math.stackexchange.com/questions/71031/a-question-about-the-additive-group-of-a-finitely-generated-integral-domain

Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I mean an element $e\ne 0$ such that for every positive integer $n$ there is an element f such that $e=nf$.

As Darji points out, another way to ask the question is this: Suppose $e\in R$ has the property that for all positive integers $n$, $e$ is congruent to an integer mod $nR$. Must $e$ be an integer?

Note: I previously posted this to Math StackExchange here: http://math.stackexchange.com/questions/71031/a-question-about-the-additive-group-of-a-finitely-generated-integral-domain

TO SUMMARIZE: Qing Liu showed that in fact any rational in $R$ determines a divisible element of $(R,+)/(\mathbb{Z},+)$, and Wilberd van der Kallen showed that all divisible elements arise in this way. I wish I could accept both answers.