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  • Let $R$ be a semi-ring. Let the right cancellation property hold in $R$. Show that if $Ra=R$ for some $a\in R$ then show that $R$ has an idempotent.

Since $a\in R\implies \exists c\in R$ such that $ca=a\implies c^2a=ca\implies c^2=c\implies c$ is idempotent.

Question:

  • Let $R$ be a semi-ring. Let the right cancellation property hold in $R$. Show that if $aR=R$ for some $a\in R$ then show that $R$ has an idempotent.

Since $a\in R\implies \exists c\in R$ such that $ac=a$.

But I can't proceed further. Please give some hints to proceed.I am wondering if it is true.

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    I assume your definition of semiring does not include the existence of $0$? – 2017-02-01
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    Yes you are right ;$(R,+)$ is a commutative semigroup and $(R,.)$ is a semigroup – 2017-02-01
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    By not reading correctly, my fault :p – 2017-02-01
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    Where did you find this question, maybe they meant left cancellation property in the second statement? I'm not really into semi-ring theory and don't know too many interesting examples to potentially find a counterexample. – 2017-02-01
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    if it is so then we have nothing to do@Mathematician42 – 2017-02-01

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