Let R=Z[√(-5)] and let I={a+b √(-5) |a, b ∈Z, a-b is even }. Show that I is a maximal ideal of R

step 1 I have given another relation here we have mod of the negative 4 over z equals to find the maxim

Let R=Z[√(-5)] and let I={a+b √(-5) |a, b ∈Z, a-b is even }....

Key Concepts

Quotient Ring and Homomorphism

The construction of a quotient ring involves partitioning a ring into cosets with respect to an ideal. A ring homomorphism with a particular ideal as its kernel effectively 'collapses' the ring along that ideal, and if the quotient ring (the image) is a field, this demonstrates that the ideal is maximal. This method is a central technique in ring theory for analyzing the structure of rings and their ideals.

Maximal Ideal

A maximal ideal is a proper ideal that is not contained within any other proper ideal in the ring. One of its key characteristics is that when you take the quotient of the ring by a maximal ideal, the result is a field. This property is instrumental in many proofs, as it provides a clear criterion for maximality.

Ideal in a Ring

An ideal is a subset of a ring that is closed under addition and under multiplication by any element from the ring. Ideals are crucial for constructing quotient rings and understanding the overall structure of a ring, serving as the building blocks for many algebraic constructions.

Quadratic Integer Ring

A quadratic integer ring, like Z[?(-5)], consists of numbers of the form a + b?d with d fixed (here d = -5) and a, b belonging to the integers. This structure is fundamental in algebraic number theory and provides a rich context for exploring properties such as factorization and the behavior of ideals.

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