Let R be right Noetherian and let S be any ring generated by...

Let $R$ be right Noetherian and let $S$ be any ring generated by $R$ and some element $y$ such that $R+R y=R+y R$. Prove that $S$ is right Noetherian. In particular, observe that this applies with $S=R[x ; \delta]$.

Key Concepts

Ore Extensions

An Ore extension is a specific type of ring extension where a new indeterminate is added with a twisting relation, typically associated with an endomorphism or a derivation. The case S = R[x; ?] mentioned in the problem is an example of an Ore extension where the derivation ? defines how the indeterminate x commutes with elements from R. Ore extensions are important in noncommutative algebra because they generalize the construction of polynomial rings while accommodating noncommutativity in a controlled way.

Right Noetherian Rings

A right Noetherian ring is a ring in which every right ideal is finitely generated. This property is essential in many areas of algebra because it guarantees that ascending chains of right ideals eventually stabilize, ensuring a form of finiteness that makes many structural arguments work. It is a fundamental concept in ring theory and module theory, as it limits the possible complexity of the ring's internal structure.

Ring Extensions by Adjoining an Element

Adjoining an element to a ring to form a new ring extension is a common construction in algebra. In this context, one starts with a ring R and extends it by adding a new element y, generating a larger ring S in which every element can be represented in terms of R and y. The specific condition R + R y = R + y R provides a kind of symmetry in the way y interacts with R, which is instrumental in preserving properties like the Noetherian condition in the extension.

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