I am at a very initial stage of commutative algebra. I want to know whether the power of a prime ideal in a commutative ring is prime ideal or not?

Yesenia Obrien
2022-07-07
Answered

I am at a very initial stage of commutative algebra. I want to know whether the power of a prime ideal in a commutative ring is prime ideal or not?

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Answered 2022-07-08
Author has **11** answers

An ideal $I\subset R$ is prime iff $R/I$ is an integral domain (that is, it contains no nontrivial zero divisors).

If $I$ is a prime ideal such that ${I}^{2}\ne I$, then in the quotient ring $R/{I}^{2}$ the elements of the form $x+{I}^{2}$, where $x\in I$, will be nilpotents. Thus, $R/{I}^{2}$ is not a domain and ${I}^{2}$ is not prime.

For a concrete and illustrative example, consider $I=(p)\subset \mathbb{Z}$ - the ideal of the ring of integers generated by a prime $p$. Then, ${I}^{2}=({p}^{2})$, and is not prime (since it is generated by a non-prime integer).

If $I$ is a prime ideal such that ${I}^{2}\ne I$, then in the quotient ring $R/{I}^{2}$ the elements of the form $x+{I}^{2}$, where $x\in I$, will be nilpotents. Thus, $R/{I}^{2}$ is not a domain and ${I}^{2}$ is not prime.

For a concrete and illustrative example, consider $I=(p)\subset \mathbb{Z}$ - the ideal of the ring of integers generated by a prime $p$. Then, ${I}^{2}=({p}^{2})$, and is not prime (since it is generated by a non-prime integer).

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Physical meaning of the null space of a matrix

What is an intuitive meaning of the null space of a matrix? Why is it useful?

I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it.

E.g.: I think of the rank r of a matrix as the minimum number of dimensions that a linear combination of its columns would have; it tells me that, if I combined the vectors in its columns in some order, I'd get a set of coordinates for an r-dimensional space, where r is minimum (please correct me if I'm wrong). So that means I can relate rank (and also dimension) to actual coordinate systems, and so it makes sense to me. But I can't think of any physical meaning for a null space... could someone explain what its meaning would be, for example, in a coordinate system?

What is an intuitive meaning of the null space of a matrix? Why is it useful?

I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it.

E.g.: I think of the rank r of a matrix as the minimum number of dimensions that a linear combination of its columns would have; it tells me that, if I combined the vectors in its columns in some order, I'd get a set of coordinates for an r-dimensional space, where r is minimum (please correct me if I'm wrong). So that means I can relate rank (and also dimension) to actual coordinate systems, and so it makes sense to me. But I can't think of any physical meaning for a null space... could someone explain what its meaning would be, for example, in a coordinate system?

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Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).

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$T([1,0,1])=[2,3],\text{}\text{}\text{}T([2,1,3])=[-1,0],\text{}\text{}\text{}T([0,0,1])=[3,7]$

Find $\text{}\text{}T([x,y,z])$

$T([1,0,1])=[2,3],\text{}\text{}\text{}T([2,1,3])=[-1,0],\text{}\text{}\text{}T([0,0,1])=[3,7]$

Find $\text{}\text{}T([x,y,z])$

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