Examples of Finite Non-Commutative Rings

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SUMMARY

This discussion provides examples of finite non-commutative rings, specifically highlighting the set of all nxn matrices over the finite field F_p for some prime p, which serves as a finite non-commutative ring with identity for n greater than 1. Additionally, participants suggest exploring subrings of this matrix ring to identify finite non-commutative rings without identity. The conversation emphasizes the importance of selecting appropriate values for n and p to generate relevant examples.

PREREQUISITES
  • Understanding of finite fields, specifically F_p.
  • Knowledge of matrix algebra, particularly nxn matrices.
  • Familiarity with ring theory concepts, including identities and non-commutativity.
  • Basic grasp of algebraic structures and their properties.
NEXT STEPS
  • Research the properties of finite fields and their applications in ring theory.
  • Study the structure of matrix rings and their identities in algebra.
  • Explore examples of subrings within matrix rings to identify non-commutative rings without identity.
  • Investigate additional examples of finite non-commutative rings in advanced algebra texts.
USEFUL FOR

Mathematicians, algebra students, and educators interested in advanced ring theory and its applications in finite fields and matrix algebra.

murshid_islam
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can anyone give me examples of:

1.finite non-commutative ring with identity
2.finite non-commutative ring without identity
 
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1. Set of all nxn matrices over F_p for some p, with respect to normal + and * (for n>1).

2. I think you should be able to find some subring of (1) that will do. I'll let you toy with it.
I'd try setting an n, and maybe also a p, if I were you.
 
Murshid,

You already have a version of this thread which has been relocated to the Homework Help section of this site. If you want to receive help on questions such as these, you should post there and before that read the sticky at the top of that Forum.

https://www.physicsforums.com/showthread.php?t=4825
 
Last edited:

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