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Ring Homomorphisms and Isomorphisms

In: Ring Theory

Author

Listed:
  • Dinesh Khattar

    (University of Delhi, Department of Mathematics)

  • Neha Agrawal

    (University of Delhi, Department of Mathematics)

Abstract

It is time to further enhance our understanding of rings by introducing a pair of new concepts: homomorphisms and isomorphisms of rings. Once again, our journey deeper into ring theory shall pass through the borderlands of groups. Why? Because the homomorphisms and isomorphisms of rings are exactly analogous to the notions of homomorphisms and isomorphisms in groups! Only a little tweaking is required to move the concept from groups to rings. Homomorphism between two groups is a mapping which preserves the binary operation. Since there are two binary operations in a ring, we define a homomorphism between two rings which preserves both the binary operations. In other words, a ring homomorphism is a structure-preserving function between two rings. So, just as Group theory requires us to look at maps which “preserve the operation”, Ring theory demands that we look at maps which preserve both operations.

Suggested Citation

  • Dinesh Khattar & Neha Agrawal, 2023. "Ring Homomorphisms and Isomorphisms," Springer Books, in: Ring Theory, chapter 0, pages 137-183, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-29440-2_4
    DOI: 10.1007/978-3-031-29440-2_4
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