SUMMARY
The discussion focuses on identifying all ideals in the ring RxR, where R represents the real numbers. The four identified ideals are (R,0), (0,R), (R,R), and (0,0). Each of these sets satisfies the conditions of being an ideal, specifically being a subgroup under addition. Additionally, the sub-ring T={(100a,100b)|a,b are elements in R} is noted as a non-ideal due to failing the ideal property when multiplied by certain elements.
PREREQUISITES
- Understanding of ring theory and ideals
- Familiarity with subgroup properties in algebra
- Knowledge of the structure of the Cartesian product of sets
- Basic concepts of real numbers and their operations
NEXT STEPS
- Study the properties of ideals in ring theory
- Explore subgroup criteria in algebraic structures
- Learn about the Cartesian product and its applications in algebra
- Investigate examples of non-ideal sub-rings in RxR
USEFUL FOR
Mathematicians, algebra students, and anyone studying ring theory and ideal structures in abstract algebra will benefit from this discussion.