What is finitely generated ideal?
An ideal in a ring is called finitely generated if and only if it can be generated by a finite set. An ideal is called principal if and only if it can be generated by a single element.
Is the ideal a principal ideal?
For a ∈ R, (a) := Ra = {ra : r ∈ R} is an ideal. An ideal of the form (a) is called a principal ideal with generator a.
What is the ideal generated by two elements?
By definition, (r)+(s) is the sum of the ideals generated by r and s. That these two are equal is actually an easy theorem. More generally, if A and B are subsets of R then (A,B) or ⟨A,B⟩ denotes the smallest ideal of R containing both A and B. If I and J are two ideals of R, then I+J denotes the set {i+j:i∈I,j∈J}.
Is 2Z a principal ideal?
The ideal 2Z of Z is the principal ideal < 2 >. Example 4 above (the polynomials in R[x] with 0 constant term) is the principal ideal < x > . The set of all polynomials in Z[x] whose coefficients are all even is the principal ideal < 2 >.
Are all ideals finitely generated?
Definition An ideal I of a ring R is finitely generated if there is a finite subset A of R such that I = 〈A〉. Example Every principal ideal is finitely generated. Theorem A ring R is Noetherian if and only if every ideal of R is finitely generated.
What is a generated ideal?
Definition 1 The ideal generated by S is the smallest ideal of R containing S, that is, the intersection of all ideals containing S.
What does it mean if an ideal is principal?
An ideal of a ring is called principal if there is an element of such that. In other words, the ideal is generated by the element . For example, the ideals of the ring of integers are all principal, and in fact all ideals of. are principal.
Is every ideal finitely generated?
In a principal ideal ring R, every left or right ideal is generated by a single element and hence in particular, it is finitely generated. Thus R is a Noetherian ring by the Theorem 1.4.
What is proper ideal?
Any ideal of a ring which is strictly smaller than the whole ring. For example, is a proper ideal of the ring of integers , since . The ideal of the polynomial ring is also proper, since it consists of all multiples of. , and the constant polynomial 1 is certainly not among them.
Is prime ideal of Z?
(1) The prime ideals of Z are (0),(2),(3),(5),…; these are all maximal except (0). (2) If A = C[x], the polynomial ring in one variable over C then the prime ideals are (0) and (x − λ) for each λ ∈ C; again these are all maximal except (0).
Is nZ an ideal of Z?
For an integer n ∈ Z, we define a subset nZ ⊆ Z as nZ = {kn | k ∈ Z}; that is, nZ consists of the multiples of n. Another notation for the set nZ is (n). (1) Prove that nZ is an ideal for every n ∈ Z.