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Soft congruence relations over rings.

Xiaolong Xin1, Wenting Li1

  • 1Department of Mathematics, Northwest University, Xi'an 710127, China.

Thescientificworldjournal
|June 21, 2014
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Summary
This summary is machine-generated.

This study introduces soft congruence relations using soft set theory to handle uncertainty. It establishes key correspondences and proves the first three soft isomorphism theorems for rings.

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Area of Science:

  • Abstract algebra
  • Set theory
  • Fuzzy mathematics

Background:

  • Soft sets provide a framework for modeling uncertainty.
  • Existing algebraic structures lack robust methods for handling imprecise data.
  • The study builds upon Molodtsov's soft set theory.

Purpose of the Study:

  • To introduce and investigate soft congruence relations within the framework of soft set theory.
  • To extend algebraic concepts like quotient rings and ideals to the soft set context.
  • To establish fundamental isomorphism theorems for soft algebraic structures.

Main Methods:

  • Application of soft set theory to define congruence relations on algebraic structures.
  • Development of novel concepts: soft quotient rings, generalized soft ideals, and generalized soft quotient rings.
  • Investigation of properties and relationships between these new soft structures.

Main Results:

  • Established one-to-one correspondences between soft congruence relations and idealistic soft rings.
  • Demonstrated a one-to-one correspondence between soft congruence relations and soft ideals.
  • Successfully proved the first, second, and third soft isomorphism theorems for rings.

Conclusions:

  • Soft congruence relations offer a powerful tool for algebraic manipulation under uncertainty.
  • The established correspondences and isomorphism theorems provide a solid foundation for soft algebraic theory.
  • This work paves the way for further research in soft algebraic structures and their applications.