Show that $R$ can be written as a direct product of two or more (nonzero) rings iff $R$ contains a non-trivial idempotent. Show that if $e$ is an idempotent, then $R=Re\times R(1-e)$ and that $Re$ may be realized as a localization, $Re=R[{e}^{-1}]$.

Alaina Marshall
2022-05-24
Answered

Show that $R$ can be written as a direct product of two or more (nonzero) rings iff $R$ contains a non-trivial idempotent. Show that if $e$ is an idempotent, then $R=Re\times R(1-e)$ and that $Re$ may be realized as a localization, $Re=R[{e}^{-1}]$.

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nicoupsqb

Answered 2022-05-25
Author has **5** answers

Assume $A\cong {A}_{1}\oplus {A}_{2}$. It follows $e:=(1,0)\ne 1$ is a non trivial idempotent. Assume conversely that $1\ne e\in A$ is a non trivial idempotent. Let $I:=(e),J:=(e-1)\subseteq A$ be the ideal generated by $e,e-1$. It follows $I+J=(1)$ hence these ideals are coprime. Moreover $IJ=I\cap J$. Since $IJ:=A(e(e-1))=A({e}^{2}-e)=A(0)=(0)$ is the zero ideal, we get by the CRT an isomorphism is the zero ideal, we get by the CRT an isomorphism

$A\cong A/IJ\cong A/I\times A/J$

where $A/I,A/J\ne 0$. Hence by the CRT we get that $A$ has a non-trivial idempotent iff $A$ may be written as a (non-trivial) direct sum of two rings.

$A\cong A/IJ\cong A/I\times A/J$

where $A/I,A/J\ne 0$. Hence by the CRT we get that $A$ has a non-trivial idempotent iff $A$ may be written as a (non-trivial) direct sum of two rings.

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Thank you so much!

$a+b+c=4$

${a}^{2}+{b}^{2}+{c}^{2}=6$

${a}^{3}+{b}^{3}+{c}^{3}=10$

Thank you so much!

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2x +2y -z =-6

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a) Provide a coefficient matrix corresponding to the system oflinear equations

b) what is the inverse of this matrix?

c) What is the transpose of the matrix?

d) find the determinant for this matrix?

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