Show that A={(3 x, y) |x, y ∈Z} is a maximal ideal of Z ⊕Z. Generalize. What happens if 3 x is r

step 1 In this question it is given that Z1, Z2, Z3 are the points on the circle of radius 5 and the ce

Show that A={(3 x, y) |x, y ∈Z} is a maximal ideal of Z ⊕Z....

Key Concepts

Quotient Ring

A quotient ring is formed by taking a ring and ‘modding out’ by an ideal. The operations on the quotient are induced naturally from the operations in the original ring. The structure of the quotient ring often reveals important properties about the ideal and the ring itself, in particular, if the quotient is a field then the ideal must be maximal.

Maximal Ideal

A maximal ideal is an ideal that is proper (not equal to the whole ring) and is not contained in any other proper ideal. The importance of a maximal ideal lies in the fact that the quotient of a ring by a maximal ideal yields a field. This relationship is a key tool in many areas of algebra, as fields have a well-understood structure and simpler properties.

Ring Homomorphism and Kernel

A ring homomorphism is a function between rings that respects both addition and multiplication. The kernel of a ring homomorphism, which is the set of elements mapping to the zero element of the codomain, is always an ideal. This concept is central to understanding how ideals can come from natural maps, and why such ideals are often maximal if the corresponding homomorphism is surjective onto a field.

Direct Sum (or Direct Product) of Rings

This concept refers to constructing a new ring from two (or more) rings by forming ordered pairs (or tuples) with component?wise addition and multiplication. It is important because properties of the individual rings can be studied separately and then ‘glued together’ to understand the direct sum, and many constructions in ring theory (and module theory) use this idea.

Ideal in a Ring

An ideal is a special subset of a ring that is closed under addition and absorbs multiplication by arbitrary ring elements. Ideals are fundamental in ring theory as they allow the construction of quotient rings, which in turn are used to study factorization properties and homomorphic images of the original ring.

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