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Zello
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A conjecturequestion on finite commutative rings

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I have considered about Lehmer's conjecture that φ(n)|(n-1) if and only if n is a prime. I generalized the conjecture onto genernal finite commutative ring. The idea is below.

 Set A is a finite commutative ring, and B is its unit group. Then there is a partition of A by B in multipication. Choose the representatives {a1,a2,...ak}, then define a subgroup Bi of B as Bi={b is belong to B, b*ai=ai}, for 1=<i<=k. Clearly, sum<1 to k>[B:Bi]=|A|-1, for {ai} is the collection of representatives. With a little more consideration, this converts to the well-known result in basic number theory that sum<d|n>φ(d)=n (using Chinese residue theorem, or fact that every artin ring is a direct sum of local rings). Now, what I conjectured is, if |B| can divide |A|, then A must be a field.
 I don't know much about finite commutative rings, but I can see its significance in number theory here. Moreover, I got to know some analytic methods used in the attacking of Lehmer's conjecture, such as Kevin Ford from UIUC. When faced the problem about rings, some representation theory may work. I don't know.

Set $A$ is a finite commutative ring, and $B$ is its unit group. Then there is a partition of $A$ by $B$ in multipication. Choose the representatives $\lbrace a_1,a_2,...a_k\rbrace$, then define a subgroup $B_i$ of $B$ as $B_i=\lbrace b \in B \mid b\cdot a_i=a_i\rbrace$, for $1 \le i \le k$. Clearly, $\sum_{i=1}^k [B:B_i]=|A|-1$, for $\lbrace a_i \rbrace$ is the collection of representatives. With a little more consideration, this converts to the well-known result in basic number theory that $\sum_{d|n}\phi(d)=n$ (using Chinese residue theorem, or fact that every artin ring is a direct sum of local rings). Now, what I conjectured is, if $|B|$ can divide $|A|$, then $A$ must be a field.

I don't know much about finite commutative rings, but I can see its significance in number theory here. Moreover, I got to know some analytic methods used in the attacking of Lehmer's conjecture, such as Kevin Ford from UIUC. When faced the problem about rings, some representation theory may work. I don't know.

I have considered about Lehmer's conjecture that φ(n)|(n-1) if and only if n is a prime. I generalized the conjecture onto genernal finite commutative ring. The idea is below.

 Set A is a finite commutative ring, and B is its unit group. Then there is a partition of A by B in multipication. Choose the representatives {a1,a2,...ak}, then define a subgroup Bi of B as Bi={b is belong to B, b*ai=ai}, for 1=<i<=k. Clearly, sum<1 to k>[B:Bi]=|A|-1, for {ai} is the collection of representatives. With a little more consideration, this converts to the well-known result in basic number theory that sum<d|n>φ(d)=n (using Chinese residue theorem, or fact that every artin ring is a direct sum of local rings). Now, what I conjectured is, if |B| can divide |A|, then A must be a field.
 I don't know much about finite commutative rings, but I can see its significance in number theory here. Moreover, I got to know some analytic methods used in the attacking of Lehmer's conjecture, such as Kevin Ford from UIUC. When faced the problem about rings, some representation theory may work. I don't know.

I have considered about Lehmer's conjecture that φ(n)|(n-1) if and only if n is a prime. I generalized the conjecture onto genernal finite commutative ring. The idea is below.

Set $A$ is a finite commutative ring, and $B$ is its unit group. Then there is a partition of $A$ by $B$ in multipication. Choose the representatives $\lbrace a_1,a_2,...a_k\rbrace$, then define a subgroup $B_i$ of $B$ as $B_i=\lbrace b \in B \mid b\cdot a_i=a_i\rbrace$, for $1 \le i \le k$. Clearly, $\sum_{i=1}^k [B:B_i]=|A|-1$, for $\lbrace a_i \rbrace$ is the collection of representatives. With a little more consideration, this converts to the well-known result in basic number theory that $\sum_{d|n}\phi(d)=n$ (using Chinese residue theorem, or fact that every artin ring is a direct sum of local rings). Now, what I conjectured is, if $|B|$ can divide $|A|$, then $A$ must be a field.

I don't know much about finite commutative rings, but I can see its significance in number theory here. Moreover, I got to know some analytic methods used in the attacking of Lehmer's conjecture, such as Kevin Ford from UIUC. When faced the problem about rings, some representation theory may work. I don't know.

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Zello
Zello
  • 55
  • 6

A conjecture on finite commutative rings

I have considered about Lehmer's conjecture that φ(n)|(n-1) if and only if n is a prime. I generalized the conjecture onto genernal finite commutative ring. The idea is below.

 Set A is a finite commutative ring, and B is its unit group. Then there is a partition of A by B in multipication. Choose the representatives {a1,a2,...ak}, then define a subgroup Bi of B as Bi={b is belong to B, b*ai=ai}, for 1=<i<=k. Clearly, sum<1 to k>[B:Bi]=|A|-1, for {ai} is the collection of representatives. With a little more consideration, this converts to the well-known result in basic number theory that sum<d|n>φ(d)=n (using Chinese residue theorem, or fact that every artin ring is a direct sum of local rings). Now, what I conjectured is, if |B| can divide |A|, then A must be a field.
 I don't know much about finite commutative rings, but I can see its significance in number theory here. Moreover, I got to know some analytic methods used in the attacking of Lehmer's conjecture, such as Kevin Ford from UIUC. When faced the problem about rings, some representation theory may work. I don't know.