Is a ring Injective over itself?

Self-injective rings Every ring with unity is a free module and hence is a projective as a module over itself, but it is rarer for a ring to be injective as a module over itself, (Lam 1999, §3B).

Why are Noetherian rings important?

One big reason why they are important is that if R is noetherian the R[X] is also noetherian which then helps us see that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in n …

What does Injective mean in math?

one-to-one function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements to distinct elements; that is, f(x1) = f(x2) implies x1 = x2. In other words, every element of the function’s codomain is the image of at most one element of its domain.

Is a principal ideal domain?

In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. All Euclidean domains and all fields are principal ideal domains.

Is a subring of a Noetherian ring Noetherian?

Quotient rings and finite direct sums of Noetherian rings are again Noetherian, but a subring of a Noetherian ring need not be Noetherian. For example, a polynomial ring in infinitely many variables over a field is not Noetherian, although it is contained in its field of fractions, which is Noetherian.

Are the integers a Noetherian ring?

Any principal ideal ring, such as the integers, is Noetherian since every ideal is generated by a single element. This includes principal ideal domains and Euclidean domains.

What is Noetherian R-module?

In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules.

How do you find the injective function?

To show that a function is injective, we assume that there are elements a1 and a2 of A with f(a1) = f(a2) and then show that a1 = a2. Graphically speaking, if a horizontal line cuts the curve representing the function at most once then the function is injective.