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slightly generalized the corollary I'm asking for a reference to
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Salvo Tringali
Salvo Tringali
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I'm looking for a reference to theThe following corollary, which comes as a by-product of a more generalabstract result, and I'm essentially looking for a reference to it (or to something more general than it).

Corollary. Let $R$ be a non-trivial Dedekind-finite unital ring (either commutative or not), $k$ a positive integer, and let $k \in \mathbf N^+$$\Gamma$ a non-trivial submonoid of $(\mathbf N^k, +)$. Then the monoid ring $R[\mathbb N^k]$$R[\Gamma]$ has infinitely many pairwise non-associate irreducible elements.

Notes. (i) A unital ring is called Dedekind-finite provided that $xy = 1_R$ for some $x, y \in R$ only if $yx = 1_R$. (ii) $\mathbb N^k$ is the monoidIf $(\mathbf N^k, +)$. So in particular, if$\Gamma = (\mathbf N^k, +)$ and $R$ is commutative, then $R[\mathbb N^k]$$R[\Gamma]$ is ajust the usual ring of polynomials in $k$ variables $X_1, \ldots, X_k$ with coefficients in $R$.

I'm looking for a reference to the following corollary, which comes as a by-product of a more general result.

Corollary. Let $R$ be a non-trivial Dedekind-finite unital ring (either commutative or not), and let $k \in \mathbf N^+$. Then the monoid ring $R[\mathbb N^k]$ has infinitely many pairwise non-associate irreducible elements.

Notes. (i) A unital ring is called Dedekind-finite provided that $xy = 1_R$ for some $x, y \in R$ only if $yx = 1_R$. (ii) $\mathbb N^k$ is the monoid $(\mathbf N^k, +)$. So in particular, if $R$ is commutative, then $R[\mathbb N^k]$ is a ring of polynomials in $k$ variables with coefficients in $R$.

The following comes as a by-product of a more abstract result, and I'm essentially looking for a reference to it (or to something more general than it).

Corollary. Let $R$ be a non-trivial Dedekind-finite unital ring (either commutative or not), $k$ a positive integer, and $\Gamma$ a non-trivial submonoid of $(\mathbf N^k, +)$. Then the monoid ring $R[\Gamma]$ has infinitely many pairwise non-associate irreducible elements.

Notes. (i) A unital ring is called Dedekind-finite provided that $xy = 1_R$ for some $x, y \in R$ only if $yx = 1_R$. (ii) If $\Gamma = (\mathbf N^k, +)$ and $R$ is commutative, then $R[\Gamma]$ is just the usual ring of polynomials in $k$ variables $X_1, \ldots, X_k$ with coefficients in $R$.

Forgot to mention that the ring $R$ is non-trivial (otherwise the claim is false)
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Salvo Tringali
Salvo Tringali
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I'm looking for a reference to the following corollary, which comes as a by-product of a more general result.

Corollary. Let $R$ be a non-trivial Dedekind-finite unital ring (either commutative or not), and let $k \in \mathbf N^+$. Then the monoid ring $R[\mathbb N^k]$ has infinitely many pairwise non-associate irreducible elements.

Notes. (i) A unital ring is called Dedekind-finite provided that $xy = 1_R$ for some $x, y \in R$ only if $yx = 1_R$. (ii) $\mathbb N^k$ is the monoid $(\mathbf N^k, +)$. So in particular, if $R$ is commutative, then $R[\mathbb N^k]$ is a ring of polynomials in $k$ variables with coefficients in $R$.

I'm looking for a reference to the following corollary, which comes as a by-product of a more general result.

Corollary. Let $R$ be a Dedekind-finite unital ring (either commutative or not), and let $k \in \mathbf N^+$. Then the monoid ring $R[\mathbb N^k]$ has infinitely many pairwise non-associate irreducible elements.

Notes. (i) A unital ring is called Dedekind-finite provided that $xy = 1_R$ for some $x, y \in R$ only if $yx = 1_R$. (ii) $\mathbb N^k$ is the monoid $(\mathbf N^k, +)$. So in particular, if $R$ is commutative, then $R[\mathbb N^k]$ is a ring of polynomials in $k$ variables with coefficients in $R$.

I'm looking for a reference to the following corollary, which comes as a by-product of a more general result.

Corollary. Let $R$ be a non-trivial Dedekind-finite unital ring (either commutative or not), and let $k \in \mathbf N^+$. Then the monoid ring $R[\mathbb N^k]$ has infinitely many pairwise non-associate irreducible elements.

Notes. (i) A unital ring is called Dedekind-finite provided that $xy = 1_R$ for some $x, y \in R$ only if $yx = 1_R$. (ii) $\mathbb N^k$ is the monoid $(\mathbf N^k, +)$. So in particular, if $R$ is commutative, then $R[\mathbb N^k]$ is a ring of polynomials in $k$ variables with coefficients in $R$.

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Salvo Tringali
Salvo Tringali
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Existence of infinitely many pairwise non-associate atoms in a ring of polynomials in $k$ variables over a Dedekind-finite unital ring

I'm looking for a reference to the following corollary, which comes as a by-product of a more general result.

Corollary. Let $R$ be a Dedekind-finite unital ring (either commutative or not), and let $k \in \mathbf N^+$. Then the monoid ring $R[\mathbb N^k]$ has infinitely many pairwise non-associate irreducible elements.

Notes. (i) A unital ring is called Dedekind-finite provided that $xy = 1_R$ for some $x, y \in R$ only if $yx = 1_R$. (ii) $\mathbb N^k$ is the monoid $(\mathbf N^k, +)$. So in particular, if $R$ is commutative, then $R[\mathbb N^k]$ is a ring of polynomials in $k$ variables with coefficients in $R$.