Need help about Quotient Rings (Factor Rings)

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SUMMARY

The discussion focuses on verifying the multiplication operation in the Factor Ring R/S, specifically demonstrating that (S + a)(S + b) = (S + ab) for an ideal S of a ring R. The user seeks to show that (S + ab) ≤ (S + a)(S + b) using the definition provided in I.N. Herstein's book. A concrete example is suggested, using S = 5Z with a = 2 and b = 3, to illustrate the concept effectively.

PREREQUISITES
  • Understanding of ring theory and ideals
  • Familiarity with Factor Rings and their properties
  • Knowledge of multiplication operations in algebraic structures
  • Basic experience with examples in modular arithmetic
NEXT STEPS
  • Study the properties of Factor Rings in detail
  • Explore I.N. Herstein's "Topics in Algebra" for deeper insights
  • Practice with additional examples of ideals and their corresponding Factor Rings
  • Learn about the relationship between ideals and homomorphisms in ring theory
USEFUL FOR

Mathematics students, algebra enthusiasts, and anyone studying abstract algebra, particularly those focusing on ring theory and Factor Rings.

AAQIB IQBAL
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Suppose S is an Ideal of a Ring R
I want to verify the multiplication operation in the Factor Ring R/S which is
(S + a)(S + b) = (S + ab)
for this i Need to show that :
(S + ab) ≤ (S +a)(S + b)
please give me some idea about it
IT IS GIVEN AS DEFINITION IN THE BOOK I.N Herstein
 
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It might help to consider a concrete example such as S=5Z with a=2 and b=3.
 

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