Are Ideals of Mn(Z) Commutative?

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In summary, the conversation involves a question about proving commutativity for two ideals I and J in a ring of matrices Mn(Z). The speaker explains their approach of using elements from the ideals and matrices, but they are unsure of how to proceed. Another person suggests checking the form of the ideals and matrices to find a way to use the hypothesis.
  • #1
DukeSteve
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Hello Experts,

Again a Q and what I did, please tell me what I am doing wrong:

Given that there is a ring of matrices above Z (integers) Mn(Z) and 2 ideals I, J of this ring.

I need to prove that they are commutative: IJ = JI

What I did is that:

For all i in I and for all M in Mn(Z) n is the the size of a matrix n x n

M*i in I and i*M is also in I.

same with J : j*M in J and M*j is in J

For every k in J and for every h in I:

kh = j*M*i*M = j*(M*i)*M = ... I don't know what to do from here... please guide me.
 
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  • #2
Hey Duke!
I suggest you to check again your hypothesys instead.
Well, [tex]\mathbb{Z}[/tex] is a ring with unity, right? What's the form of it's ideals? What's the form of the ideals of [tex]M_{n}(\mathbb{Z})[/tex]?

The problem of your approach is that I can't really see a way to use your hypothesys.
 

FAQ: Are Ideals of Mn(Z) Commutative?

What is the ideal of a ring?

The ideal of a ring is a subset of the ring that satisfies certain properties. It is a special type of subring that has additional properties such as absorbing elements and closure under multiplication by elements from the ring.

How is an ideal different from a subring?

An ideal is a special type of subring that has additional properties, such as absorbing elements and closure under multiplication by elements from the ring. A subring, on the other hand, does not necessarily have these properties.

Can an ideal be empty?

Yes, an ideal can be empty. An ideal is a subset of a ring, and if the ring itself is empty, then any subset of it, including the ideal, will also be empty.

What is the significance of ideals in ring theory?

Ideals play a crucial role in ring theory as they help us understand the structure and properties of a ring. They allow us to define quotient rings, which are useful in many mathematical contexts, such as algebraic geometry and number theory.

How can I determine if a subset of a ring is an ideal?

To determine if a subset of a ring is an ideal, you can check if it satisfies the two defining properties of an ideal: closure under addition and multiplication by elements from the ring. Additionally, an ideal must also contain the additive identity element of the ring.

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