The sum of two non-units of a ring $R$ is a non-unit implies that the Jacobson radical is maximal.
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My problem is: If the sum of to non-units in a ring $R$ is non-unit, then the Jacobson radical $J(R)$ is maximal. I need help please. I have no idea how to start.
I thought that if the set of all non-units is an ideal of $R$, then it turns to be the unique maximal left ideal of $R$ and so it is $J(R)$.
noncommutative-algebra
asked Dec 24 '18 at 21:50
Hussein EidHussein Eid
63
$endgroup$
add a comment |
$begingroup$
My problem is: If the sum of to non-units in a ring $R$ is non-unit, then the Jacobson radical $J(R)$ is maximal. I need help please. I have no idea how to start.
I thought that if the set of all non-units is an ideal of $R$, then it turns to be the unique maximal left ideal of $R$ and so it is $J(R)$.
noncommutative-algebra
asked Dec 24 '18 at 21:50
Hussein EidHussein Eid
63
$endgroup$
$begingroup$
Can you show that this implies the set of non-units is an ideal?
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:52
add a comment |
$begingroup$
My problem is: If the sum of to non-units in a ring $R$ is non-unit, then the Jacobson radical $J(R)$ is maximal. I need help please. I have no idea how to start.
I thought that if the set of all non-units is an ideal of $R$, then it turns to be the unique maximal left ideal of $R$ and so it is $J(R)$.
noncommutative-algebra
asked Dec 24 '18 at 21:50
Hussein EidHussein Eid
63
$endgroup$
My problem is: If the sum of to non-units in a ring $R$ is non-unit, then the Jacobson radical $J(R)$ is maximal. I need help please. I have no idea how to start.
I thought that if the set of all non-units is an ideal of $R$, then it turns to be the unique maximal left ideal of $R$ and so it is $J(R)$.
noncommutative-algebra
noncommutative-algebra
asked Dec 24 '18 at 21:50
Hussein EidHussein Eid
63
asked Dec 24 '18 at 21:50
Hussein EidHussein Eid
63
asked Dec 24 '18 at 21:50
Hussein EidHussein Eid
63
asked Dec 24 '18 at 21:50
Hussein EidHussein Eid
63
asked Dec 24 '18 at 21:50
Hussein EidHussein Eid
63
63
$begingroup$
Can you show that this implies the set of non-units is an ideal?
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:52
add a comment |
$begingroup$
Can you show that this implies the set of non-units is an ideal?
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:52
$begingroup$
Can you show that this implies the set of non-units is an ideal?
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:52
$begingroup$
Can you show that this implies the set of non-units is an ideal?
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:52
add a comment |
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$begingroup$
Can you show that this implies the set of non-units is an ideal?
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 21:52