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Please suggest me some reference to read the following topics of ring theory:

  1. Congruence on rings

  2. One-one correspondence between the set of ideals and the set of all congruences on a ring.

They appears in this syllabus but I could not find any material to read from.

Please help.

Added: Is Congruence on rings by any means indicating towards ideals?

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Jave
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1 Answers1

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This topic is usually omitted in first courses on Abstract Algebra, but is almost always covered in courses on Universal Algebra. For example here are links to discussions in Malcev: Algebraic Systems and McKenzie, McNulty, & Taylor: Algebras, Lattices, Varieties

Is Congruence on rings by any means indicating towards ideals?

Just as in $\Bbb Z$ a congruence on an algebraic structure is an equivalence relation '$\equiv$' that is compatible with all the structure's basic operations $f\,$ (e.g. addition and multiplication in a ring), i.e. $$ a_i\equiv b_i \Rightarrow\, f(a_1,\ldots, a_k) \equiv f(b_1,\ldots, b_k)$$

which is precisely what is needed for the operations to be well-defined on the equivalence classes.

For rings, it is easy to verify that the set $\, I_\equiv = \{ a\ :\ a\equiv 0\}$ is an ideal and, further, $\, \equiv\: \to I_\equiv\,$ yields a one-to-one correspondence between congruences and ideals, which allows us to simplify the study of such quotient structures by focusing on (simpler) ideals instead of congruences.

The algebraic structures (e.g. groups, rings, Boolean algebras) studied in most first algebra courses enjoy this simplification, i.e. their congruences are determined by a single congruence class (so-called ideal-determined varieties. Generally this simplification is not possible so we must instead work with general congruence relations when studying quotients of algebraic structures.

Note also that congruences on $R$ can be viewed as sub-$R$-algebras of the square $R^2$.

Bill Dubuque
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