If $R$ is a Boolean ring with $mid R mid > 2$, determine all the zero divisors of $R$.












2














If $R$ is a Boolean ring with $mid R mid > 2$, determine all the zero divisors of $R$.

My attempt: Let $a, bin R$, $a neq 0$ and $ab = 0$. How do I prove that $b = 0$?






abstract-algebra ring-theory







share|cite|improve this question






















  • Problem given: find zero divisors of this ring $R$. Approach chosen: suppose $b$ is a zero divisor and prove it is zero. If you look at this summary of what happened, I think you'll agree you need to think about what you are doing a little more thoroughly.
    – rschwieb
    Apr 29 '15 at 17:30


















2














If $R$ is a Boolean ring with $mid R mid > 2$, determine all the zero divisors of $R$.

My attempt: Let $a, bin R$, $a neq 0$ and $ab = 0$. How do I prove that $b = 0$?






abstract-algebra ring-theory







share|cite|improve this question






















  • Problem given: find zero divisors of this ring $R$. Approach chosen: suppose $b$ is a zero divisor and prove it is zero. If you look at this summary of what happened, I think you'll agree you need to think about what you are doing a little more thoroughly.
    – rschwieb
    Apr 29 '15 at 17:30
















2












2








2


1





If $R$ is a Boolean ring with $mid R mid > 2$, determine all the zero divisors of $R$.

My attempt: Let $a, bin R$, $a neq 0$ and $ab = 0$. How do I prove that $b = 0$?






abstract-algebra ring-theory







share|cite|improve this question













If $R$ is a Boolean ring with $mid R mid > 2$, determine all the zero divisors of $R$.

My attempt: Let $a, bin R$, $a neq 0$ and $ab = 0$. How do I prove that $b = 0$?






abstract-algebra ring-theory




abstract-algebra ring-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Apr 29 '15 at 16:13









In78

424311




424311












  • Problem given: find zero divisors of this ring $R$. Approach chosen: suppose $b$ is a zero divisor and prove it is zero. If you look at this summary of what happened, I think you'll agree you need to think about what you are doing a little more thoroughly.
    – rschwieb
    Apr 29 '15 at 17:30




















  • Problem given: find zero divisors of this ring $R$. Approach chosen: suppose $b$ is a zero divisor and prove it is zero. If you look at this summary of what happened, I think you'll agree you need to think about what you are doing a little more thoroughly.
    – rschwieb
    Apr 29 '15 at 17:30


















Problem given: find zero divisors of this ring $R$. Approach chosen: suppose $b$ is a zero divisor and prove it is zero. If you look at this summary of what happened, I think you'll agree you need to think about what you are doing a little more thoroughly.
– rschwieb
Apr 29 '15 at 17:30






Problem given: find zero divisors of this ring $R$. Approach chosen: suppose $b$ is a zero divisor and prove it is zero. If you look at this summary of what happened, I think you'll agree you need to think about what you are doing a little more thoroughly.
– rschwieb
Apr 29 '15 at 17:30












1 Answer
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Since $R$ is a Boolean ring, we have $a^2 = a$ for all $a in R$; since $vert R vert > 2$, there must exist an element $a in R$ with $a ne 0$, $a ne 1$.
Then $1 - a ne 0$, but



$a(1 - a) = a - a^2 = a - a = 0, tag{1}$



which shows that every $a in R setminus {0, 1 }$ is a zero divisor. Since $0$ and $1$ are not zero divisors, the set



$N = R setminus {0, 1 } ne emptyset tag{2}$



is precisely the set of all zero divisors of $R$.






share|cite|improve this answer





















  • Does a boolean ring always contain an identity? @Robert Lewis
    – In78
    May 1 '15 at 17:21











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1 Answer
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1 Answer
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active

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active

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4














Since $R$ is a Boolean ring, we have $a^2 = a$ for all $a in R$; since $vert R vert > 2$, there must exist an element $a in R$ with $a ne 0$, $a ne 1$.
Then $1 - a ne 0$, but



$a(1 - a) = a - a^2 = a - a = 0, tag{1}$



which shows that every $a in R setminus {0, 1 }$ is a zero divisor. Since $0$ and $1$ are not zero divisors, the set



$N = R setminus {0, 1 } ne emptyset tag{2}$



is precisely the set of all zero divisors of $R$.






share|cite|improve this answer





















  • Does a boolean ring always contain an identity? @Robert Lewis
    – In78
    May 1 '15 at 17:21
















4














Since $R$ is a Boolean ring, we have $a^2 = a$ for all $a in R$; since $vert R vert > 2$, there must exist an element $a in R$ with $a ne 0$, $a ne 1$.
Then $1 - a ne 0$, but



$a(1 - a) = a - a^2 = a - a = 0, tag{1}$



which shows that every $a in R setminus {0, 1 }$ is a zero divisor. Since $0$ and $1$ are not zero divisors, the set



$N = R setminus {0, 1 } ne emptyset tag{2}$



is precisely the set of all zero divisors of $R$.






share|cite|improve this answer





















  • Does a boolean ring always contain an identity? @Robert Lewis
    – In78
    May 1 '15 at 17:21














4












4








4






Since $R$ is a Boolean ring, we have $a^2 = a$ for all $a in R$; since $vert R vert > 2$, there must exist an element $a in R$ with $a ne 0$, $a ne 1$.
Then $1 - a ne 0$, but



$a(1 - a) = a - a^2 = a - a = 0, tag{1}$



which shows that every $a in R setminus {0, 1 }$ is a zero divisor. Since $0$ and $1$ are not zero divisors, the set



$N = R setminus {0, 1 } ne emptyset tag{2}$



is precisely the set of all zero divisors of $R$.






share|cite|improve this answer












Since $R$ is a Boolean ring, we have $a^2 = a$ for all $a in R$; since $vert R vert > 2$, there must exist an element $a in R$ with $a ne 0$, $a ne 1$.
Then $1 - a ne 0$, but



$a(1 - a) = a - a^2 = a - a = 0, tag{1}$



which shows that every $a in R setminus {0, 1 }$ is a zero divisor. Since $0$ and $1$ are not zero divisors, the set



$N = R setminus {0, 1 } ne emptyset tag{2}$



is precisely the set of all zero divisors of $R$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Apr 29 '15 at 16:26









Robert Lewis

43.2k22863




43.2k22863












  • Does a boolean ring always contain an identity? @Robert Lewis
    – In78
    May 1 '15 at 17:21


















  • Does a boolean ring always contain an identity? @Robert Lewis
    – In78
    May 1 '15 at 17:21
















Does a boolean ring always contain an identity? @Robert Lewis
– In78
May 1 '15 at 17:21




Does a boolean ring always contain an identity? @Robert Lewis
– In78
May 1 '15 at 17:21


















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