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The proof of Cantor's theorem is straightforwardly adapted to show that assuming a set of all sets U exists, then considering its Russell set R U leads to the contradiction: R U ∈ R U R U ∉ R U . {\displaystyle R_{U}\in R_{U}\iff R_{U}\notin R_{U}.}
If R is a (not necessarily commutative) Noetherian ring and I a right ideal in R, then I has a unique irredundant decomposition into tertiary ideals I = T 1 ∩ ⋯ ∩ T n {\displaystyle I=T_{1}\cap \dots \cap T_{n}} .
In the left hand sides of the following identities, L is the L eft most set and R is the R ight most set. Whenever necessary, both L and R should be assumed to be subsets of some universe set X , so that L ∁ := X ∖ L and R ∁ := X ∖ R . {\displaystyle L^{\complement }:=X\setminus L{\text{ and }}R^{\complement }:=X\setminus R.}
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Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R = K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum of T (as an operator).
Theorem — Let P a polynomial function on R n with real coefficients, F the Fourier transform considered as a unitary map L 2 (R n) → L 2 (R n). Then F * P (D) F is essentially self-adjoint and its unique self-adjoint extension is the operator of multiplication by the function P .
A ring R is called a Jacobson ring if the nilradical and Jacobson radical of R/P coincide for all prime ideals P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.
Let R be an integral domain with K its field of fractions. Then every finitely generated R-submodule I of K is a fractional ideal: that is, there is some nonzero r in R such that rI is contained in R. Indeed, one can take r to be the product of the denominators of the generators of I. If R is Noetherian, then every fractional ideal arises in ...