Return to Question

Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian.

Definition Given the dual group $\widehat{R}$ of $(R,+)$, character $\chi\in\widehat{R}$ is said to be primitive with respect to a collection $\mathcal{C}$ of subgroups of $(R,+)$, if $\chi$ does not descend to a character on any quotient group $R/S$ for all $S\in \mathcal{C}$. To descend here is canonical in the sense that $\chi$ is invariant on the elements of any coset of $R/S$.

My question is the following problem:

Are there any generic non-trivial conditions on $\mathcal{C}$ such that $$\#\{\text{primitive* characters in $\widehat{R}$ w.r.t $\mathcal{C}$}\}=\#R^\times,$$ i.e. when does the number of primitive* characters of the abelian group $(R,+)$ w.r.t $\mathcal{C}$ equal the number of units in $R$?

NOTES
The archetypal example is the ring $\mathbb{Z}/n\mathbb{Z}=(\{0,1,2,\dotsc,n-1\},+,\cdot)$ of integers modulo $n$ where the answer is positive when one considers all proper quotients.

(Added later) Qiaochu Yuan's answer shows that $\mathbb{Z}/n\mathbb{Z}$ is indeed the only example when one considers all proper subquotients. My question is whether a well-defined non-trivial collection of, rather than all, quotients leads to an affirmation of the equality above.

Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian.

My question is the following problem:

Given the dual group $\widehat{R}$ of $(R,+)$, under what conditions of $R$ is it true that

$$ \#\{\text{primitive* characters in ($\widehat{R},\cdot$)} =\#R^\times, $$ i.e. when does the number of primitive* characters of the abelian group $(R,+)$ equal the number of units in $R$?

*((Added)). A character $\chi\in\widehat{R}$ is said to be primitive with respect to a collection $\mathcal{C}$ of subgroups of $(R, +)$ if $\chi$ does not descend to a character on any quotient group $R/S$ for all $S\in\mathcal{C}$. To descend here is canonical in the sense that $\chi$ is invariant on the elements of any coset of $R/S$.

NOTES
The archetypal example is the ring $\mathbb{Z}/n\mathbb{Z}=(\{0,1,2,\dotsc,n-1\},+,\cdot)$ of integers modulo $n$ where the answer is positive when $\mathcal{C}$ contains all $\mathbb{Z}/d\mathbb{Z}$ for all $d$ divides $n$.

Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian. My question is the following problem:

Given the dual group $\widehat{R}$ of $(R,+)$, under what condition on $R$ is it true that $$\#\{\text{primitive* characters in $(\widehat{R},\cdot)$}\}=\#R^\times,$$ i.e. when does the number of primitive* characters of the abelian group $(R,+)$ equal the number of units in $R$?

*((Added)). A character $\chi\in\widehat{R}$ is said to be primitive with respect to a collection $\mathcal{C}$ of subgroups of $(R,+)$, if $\chi$ does not descend to a character on any quotient group $R/S$ for all $S\in \mathcal{C}$. To descend here is canonical in the sense that $\chi$ is invariant on the elements of any coset of $R/S$.

NOTES
The archetypal example is the ring $\mathbb{Z}/n\mathbb{Z}=(\{0,1,2,\dotsc,n-1\},+,\cdot)$ of integers modulo $n$ where the answer is positive when $\mathcal{C}$ contains all $\mathbb{Z}/d\mathbb{Z}$ for all $d$ divides $n$.

Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian.

Definition Given the dual group $\widehat{R}$ of $(R,+)$, character $\chi\in\widehat{R}$ is said to be primitive with respect to a collection $\mathcal{C}$ of subgroups of $(R,+)$, if $\chi$ does not descend to a character on any quotient group $R/S$ for all $S\in \mathcal{C}$. To descend here is canonical in the sense that $\chi$ is invariant on the elements of any coset of $R/S$.

My question is the following problem:

Are there any generic non-trivial conditions on $\mathcal{C}$ such that $$\#\{\text{primitive* characters in $\widehat{R}$ w.r.t $\mathcal{C}$}\}=\#R^\times,$$ i.e. when does the number of primitive* characters of the abelian group $(R,+)$ w.r.t $\mathcal{C}$ equal the number of units in $R$?

NOTES
The archetypal example is the ring $\mathbb{Z}/n\mathbb{Z}=(\{0,1,2,\dotsc,n-1\},+,\cdot)$ of integers modulo $n$ where the answer is positive when one considers all proper quotients.

(Added later) Qiaochu Yuan's answer shows that $\mathbb{Z}/n\mathbb{Z}$ is indeed the only example when one considers all proper subquotients. My question is whether a well-defined non-trivial collection of, rather than all, quotients leads to an affirmation of the equality above.

Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian. My question is the following problem:

Given the dual group $\widehat{R}$ of $(R,+)$, under what condition on $R$ is it true that $$\#\{\text{primitive* characters in $(\widehat{R},\cdot)$}\}=\#R^\times,$$ i.e. when does the number of primitive* characters of the abelian group $(R,+)$ equal the number of units in $R$?

*My apologies. A character $\chi\in\widehat{R}$ is said to be primitive with respect to a collection $\mathcal{C}$ of subgroups of $(R,+)$, if $\chi$ does not descend to a character on any quotient group $R/S$ for all $S\in \mathcal{C}$. To descend here means that $\chi$ is constant on all coset of $R/S$.

NOTES
The archetypal example is the ring $(\{0,1,2,\dotsc,n-1\},+,\cdot)$ of integers modulo $n$ where the answer is positive. I am not sure if the result is true for commutative rings.

Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian. My question is the following problem:

Given the dual group $\widehat{R}$ of $(R,+)$, under what condition on $R$ is it true that $$\#\{\text{primitive* characters in $(\widehat{R},\cdot)$}\}=\#R^\times,$$ i.e. when does the number of primitive* characters of the abelian group $(R,+)$ equal the number of units in $R$?

*((Added)). A character $\chi\in\widehat{R}$ is said to be primitive with respect to a collection $\mathcal{C}$ of subgroups of $(R,+)$, if $\chi$ does not descend to a character on any quotient group $R/S$ for all $S\in \mathcal{C}$. To descend here is canonical in the sense that $\chi$ is invariant on the elements of any coset of $R/S$.

NOTES
The archetypal example is the ring $\mathbb{Z}/n\mathbb{Z}=(\{0,1,2,\dotsc,n-1\},+,\cdot)$ of integers modulo $n$ where the answer is positive when $\mathcal{C}$ contains all $\mathbb{Z}/d\mathbb{Z}$ for all $d$ divides $n$.