Proving rI is a Minimal Right Ideal in R

  • Thread starter jem05
  • Start date
In summary, the conversation discusses proving that rI is a minimal right ideal of a ring R, given that I is a minimal right ideal. The individual is having trouble with the proof and asks for hints. The suggested approach is to define a new right ideal I_0 and use the properties of I and K to show that it is a right ideal.
  • #1
jem05
56
0

Homework Statement



R is a ring, r in R.
I is a minimal right ideal of R
rI =/= 0
prove rI is a minimal right ideal.


Homework Equations



rI minimal

The Attempt at a Solution



i proved rI right ideal, but I am having trouble with minimal, the thing is Ir is not a left nor a right ideal.
i tried supposing that there is a K right ideal not null and nor = rI but between them, and tried to consruct a right ideal that is neither null nor = I but also between them, to get a contradiction, but i failed.
any hint is appreciated.
thank you
 
Physics news on Phys.org
  • #2
Suppose K is a non-zero right ideal contained in rI. What about defining

[tex]I_0=\{i\in I:\, ri\in K\}[/tex]

and proving that it is a right ideal using the fact that I, K are right ideals?
 

FAQ: Proving rI is a Minimal Right Ideal in R

1. What is a minimal right ideal in a ring?

A minimal right ideal in a ring is a subset of the ring that is both a right ideal and does not contain any proper (smaller) right ideals. This means that there is no smaller subset that can also be a right ideal in the ring.

2. How do you prove that rI is a minimal right ideal in R?

To prove that rI is a minimal right ideal in R, you must show that rI is a right ideal in R and that it does not contain any proper right ideals. This can be done by showing that rI satisfies the definition of a right ideal and by using contradiction to show that there are no proper right ideals contained in rI.

3. What does it mean for rI to be a right ideal?

A right ideal is a subset of a ring that is closed under multiplication by elements on the right. This means that if a and b are elements of the right ideal, then a*b is also in the right ideal. In other words, multiplying any element in the right ideal by any element in the ring will always result in an element that is still in the right ideal.

4. What is the importance of proving that rI is a minimal right ideal in R?

Proving that rI is a minimal right ideal in R is important because it shows that rI is the smallest possible right ideal in R. This can be useful in understanding the structure of the ring and can also be used in further proofs and calculations involving the ring.

5. Can a ring have more than one minimal right ideal?

Yes, a ring can have more than one minimal right ideal. This is because there may be more than one subset of the ring that is both a right ideal and does not contain any proper right ideals. In fact, a ring can have infinitely many minimal right ideals if it is an infinite ring.

Similar threads

Union of Prime Ideals Contains an Ideal
Replies
5
Views
2K
Field extensions and radical ideals (2 problems)
Replies
10
Views
3K
A simple probability problem involving the law of total probability
Replies
1
Views
12K
How Do Primary Ideals in a PID Relate to Their Product and Intersection?
Replies
1
Views
1K
Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1
Replies
8
Views
987
Find monic generators of the ideals
Replies
1
Views
2K
Integral Domains: Homework Statement & Equations
Replies
3
Views
943
Reformulation of quadratic program as SDP program
Replies
1
Views
1K
Prove a subset of R^n is compact
Replies
4
Views
2K
Radicals and the Ring of Quotients
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top