Ring Theory: Is the ring $S$ a prime ring? Ref. Page no. 8 (Section 2) Neal H. McCoy “The Theory of...
Let $R$ be a non commutative prime ring without unity and let $S$ be the set of all ordered pair $(a,i)$, where $ain R$ and $iin I$, where $I$ is the ring of integers. On $S$ we define addition and multiplication as follows:
$(a,i)+(b,j)=(a+b,i+j)$,
$(a,i)(b,j)=(ab+ib+ja, ij)$.
Then $S$ is a ring with unity $(0,1)$.
My question-
Is the ring $S$ a prime ring?
In case $R$ is a commutative prime ring with unity, I have shown $S$ is not a prime ring.
But I am stuck in the case $R$ is non commutative prime ring without unity.
ring-theory
asked 7 hours ago
nazim khan
555
add a comment |
Let $R$ be a non commutative prime ring without unity and let $S$ be the set of all ordered pair $(a,i)$, where $ain R$ and $iin I$, where $I$ is the ring of integers. On $S$ we define addition and multiplication as follows:
$(a,i)+(b,j)=(a+b,i+j)$,
$(a,i)(b,j)=(ab+ib+ja, ij)$.
Then $S$ is a ring with unity $(0,1)$.
My question-
Is the ring $S$ a prime ring?
In case $R$ is a commutative prime ring with unity, I have shown $S$ is not a prime ring.
But I am stuck in the case $R$ is non commutative prime ring without unity.
ring-theory
asked 7 hours ago
nazim khan
555
If you have already shown the statement to be false for commutative ring with identity, what is there left to do? You have not been asked to prove it is false for all categories of rings.
– rschwieb
49 mins ago
add a comment |
Let $R$ be a non commutative prime ring without unity and let $S$ be the set of all ordered pair $(a,i)$, where $ain R$ and $iin I$, where $I$ is the ring of integers. On $S$ we define addition and multiplication as follows:
$(a,i)+(b,j)=(a+b,i+j)$,
$(a,i)(b,j)=(ab+ib+ja, ij)$.
Then $S$ is a ring with unity $(0,1)$.
My question-
Is the ring $S$ a prime ring?
In case $R$ is a commutative prime ring with unity, I have shown $S$ is not a prime ring.
But I am stuck in the case $R$ is non commutative prime ring without unity.
ring-theory
asked 7 hours ago
nazim khan
555
Let $R$ be a non commutative prime ring without unity and let $S$ be the set of all ordered pair $(a,i)$, where $ain R$ and $iin I$, where $I$ is the ring of integers. On $S$ we define addition and multiplication as follows:
$(a,i)+(b,j)=(a+b,i+j)$,
$(a,i)(b,j)=(ab+ib+ja, ij)$.
Then $S$ is a ring with unity $(0,1)$.
My question-
Is the ring $S$ a prime ring?
In case $R$ is a commutative prime ring with unity, I have shown $S$ is not a prime ring.
But I am stuck in the case $R$ is non commutative prime ring without unity.
ring-theory
ring-theory
asked 7 hours ago
nazim khan
555
asked 7 hours ago
nazim khan
555
asked 7 hours ago
nazim khan
555
asked 7 hours ago
nazim khan
555
asked 7 hours ago
nazim khan
555
555
If you have already shown the statement to be false for commutative ring with identity, what is there left to do? You have not been asked to prove it is false for all categories of rings.
– rschwieb
49 mins ago
add a comment |
If you have already shown the statement to be false for commutative ring with identity, what is there left to do? You have not been asked to prove it is false for all categories of rings.
– rschwieb
49 mins ago
If you have already shown the statement to be false for commutative ring with identity, what is there left to do? You have not been asked to prove it is false for all categories of rings.
– rschwieb
49 mins ago
If you have already shown the statement to be false for commutative ring with identity, what is there left to do? You have not been asked to prove it is false for all categories of rings.
– rschwieb
49 mins ago
add a comment |
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If you have already shown the statement to be false for commutative ring with identity, what is there left to do? You have not been asked to prove it is false for all categories of rings.
– rschwieb
49 mins ago