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Nearrings whose set of N-subgroups is linearly ordered
... Proof of Theorem 1, (a) ⇒ (c). Let N be a wd nearring with ACC on N -subgroups. Then each N -subgroup H is monogenic, that is, there exists h ∈ N such that H = N ∗ h, ([2], Theorem 5). In particular, there exists l ∈ N such that {k ∈ N | N ∗ k 6= N } = N ∗ l. Since l is an element of N ∗ l, we have ...
... Proof of Theorem 1, (a) ⇒ (c). Let N be a wd nearring with ACC on N -subgroups. Then each N -subgroup H is monogenic, that is, there exists h ∈ N such that H = N ∗ h, ([2], Theorem 5). In particular, there exists l ∈ N such that {k ∈ N | N ∗ k 6= N } = N ∗ l. Since l is an element of N ∗ l, we have ...
rings of quotients of rings of functions
... ba 6= 0, and so there exists a ∈ (ba) A for which baa 6= 0; then 0 6= aa ∈ (a) ∩ b−1 A. Theorem. Let B ⊃ A. If A is semi-prime, then B is a ring of quotients of A if and only if b · (b−1 A) 6= 0 for all nonzero b ∈ B—that is, for 0 6= b ∈ B, there exists a ∈ A such that 0 6= ba ∈ A. Proof. This is a ...
... ba 6= 0, and so there exists a ∈ (ba) A for which baa 6= 0; then 0 6= aa ∈ (a) ∩ b−1 A. Theorem. Let B ⊃ A. If A is semi-prime, then B is a ring of quotients of A if and only if b · (b−1 A) 6= 0 for all nonzero b ∈ B—that is, for 0 6= b ∈ B, there exists a ∈ A such that 0 6= ba ∈ A. Proof. This is a ...
Chapter I, Section 6
... with R Noetherian. X is Noetherian if it has a finite such covering [Liu, 2.3.45]. If X is locally Noetherian, then OX is coherent, a quasi-coherent sheaf of OX modules is coherent iff it is locally finitely generated, and every quasi-coherent subsheaf of a coherent sheaf of OX modules is coherent. ...
... with R Noetherian. X is Noetherian if it has a finite such covering [Liu, 2.3.45]. If X is locally Noetherian, then OX is coherent, a quasi-coherent sheaf of OX modules is coherent iff it is locally finitely generated, and every quasi-coherent subsheaf of a coherent sheaf of OX modules is coherent. ...
Rainbow Arithmetic Progressions in Finite Abelian Groups.
... groups with odd order is applying Lemma 1 to create a well-defined auxiliary coloring of a specific subgroup. Let G be a group and n be an odd positive integer. Partition G × Zn by letting Pg = {(g, x)|x ∈ Zn } for each g ∈ G. Without loss of generality, let |c(Pg )| ≤ |c(P0 )| for all g ∈ G. Since ...
... groups with odd order is applying Lemma 1 to create a well-defined auxiliary coloring of a specific subgroup. Let G be a group and n be an odd positive integer. Partition G × Zn by letting Pg = {(g, x)|x ∈ Zn } for each g ∈ G. Without loss of generality, let |c(Pg )| ≤ |c(P0 )| for all g ∈ G. Since ...
1 Sets and classes
... If N is a satisfies the conditions described above, and if the class of all translation of sets of N is taken for a base, then, with respect to the topology so defined, X becomes a topological group. A subset E of a topological group X is bounded if, for neighborhood U Severy n of e, there exists a ...
... If N is a satisfies the conditions described above, and if the class of all translation of sets of N is taken for a base, then, with respect to the topology so defined, X becomes a topological group. A subset E of a topological group X is bounded if, for neighborhood U Severy n of e, there exists a ...
On finite primary rings and their groups of units
... If R is an infinite primary ring, its group of units cannot be cyclic. For if 0 Nk+1 Nk, Nk is a vector space over the field R/N and thus Nk cannot be cyclic. But Nk ~ 1+Nk, a subgroup of the group G of units of R. Hence G cannot be cyclic if N ~ 0. If N 0, R is a field and it is easy to see that it ...
... If R is an infinite primary ring, its group of units cannot be cyclic. For if 0 Nk+1 Nk, Nk is a vector space over the field R/N and thus Nk cannot be cyclic. But Nk ~ 1+Nk, a subgroup of the group G of units of R. Hence G cannot be cyclic if N ~ 0. If N 0, R is a field and it is easy to see that it ...
Dynamical systems and van der Waerden`s theorem
... X is a compact metric space and T is a continuous function from X to itself. A function T : X → X is called continuous if whenever points x and y are sufficiently close to one another, the points T (x) and T (y ) can’t be too far apart. More precisely, this means that for every number > 0, there i ...
... X is a compact metric space and T is a continuous function from X to itself. A function T : X → X is called continuous if whenever points x and y are sufficiently close to one another, the points T (x) and T (y ) can’t be too far apart. More precisely, this means that for every number > 0, there i ...
congruent numbers and elliptic curves
... Remark 4.1. Projective planes can be constructed over sets other then the complex numbers. For example, P2R and P2Q are both defined analogously to P2C . The projective plane is a generalization of the ordinary xy-plane. If we set z = 1, then we regain the familiar points (x, y). This follows from t ...
... Remark 4.1. Projective planes can be constructed over sets other then the complex numbers. For example, P2R and P2Q are both defined analogously to P2C . The projective plane is a generalization of the ordinary xy-plane. If we set z = 1, then we regain the familiar points (x, y). This follows from t ...
Homomorphisms and Topological Semigroups.
... convolution multiplication, but does not give the analo gous results of the group algebras except in the case of finite semigroups. ...
... convolution multiplication, but does not give the analo gous results of the group algebras except in the case of finite semigroups. ...
Notes on Weak Topologies
... The members of τ are called open sets. Let X be a non empty set. Then τi = {{∅}, X} and τd = P(X) are topologies and are called as the indiscrete topology and the discrete topology, respectively. Note that if τ is any other topology on X, then τi ⊂ τ ⊂ τd . Let τ1 and τ2 be two topologies on X. Then ...
... The members of τ are called open sets. Let X be a non empty set. Then τi = {{∅}, X} and τd = P(X) are topologies and are called as the indiscrete topology and the discrete topology, respectively. Note that if τ is any other topology on X, then τi ⊂ τ ⊂ τd . Let τ1 and τ2 be two topologies on X. Then ...
Semantical evaluations as monadic second-order
... Monadic second-order (MS2) formulas written with inc can use quantifications on sets of edges. Existence of Hamiltonian circuit is expressible by an MS2 formula, but not by an MS formula. Theorem : MS2 formulas are no more powerful than MS formulas : for graphs of degree < d, or of tree-width < k, o ...
... Monadic second-order (MS2) formulas written with inc can use quantifications on sets of edges. Existence of Hamiltonian circuit is expressible by an MS2 formula, but not by an MS formula. Theorem : MS2 formulas are no more powerful than MS formulas : for graphs of degree < d, or of tree-width < k, o ...
Prime numbers in certain arithmetic progressions
... q ≡ 1 (mod 4 ) not in the list p1 , . . . , pk . This gives us an infinitude of such primes provided we have one. Since 5 ≡ 1 (mod 4), we have a proof for this progression. A similar proof exists for 3 (mod 4). In this case, we use the polynomial g(x) = 4x − 1 . If there are only finitely many such ...
... q ≡ 1 (mod 4 ) not in the list p1 , . . . , pk . This gives us an infinitude of such primes provided we have one. Since 5 ≡ 1 (mod 4), we have a proof for this progression. A similar proof exists for 3 (mod 4). In this case, we use the polynomial g(x) = 4x − 1 . If there are only finitely many such ...
Finite fields Michel Waldschmidt Contents
... are in Z[X]. The gcd of the coefficients of a non–zero polynomial f ∈ Z[X] is called the content of f . We denote it by c(f ). A non–zero polynomial with content 1 is called primitive. Any non–zero polynomial in Z[X] can be written in a unique way as f = c(f )g with g ∈ Z[X] primitive. For any non–z ...
... are in Z[X]. The gcd of the coefficients of a non–zero polynomial f ∈ Z[X] is called the content of f . We denote it by c(f ). A non–zero polynomial with content 1 is called primitive. Any non–zero polynomial in Z[X] can be written in a unique way as f = c(f )g with g ∈ Z[X] primitive. For any non–z ...
Derived splinters in positive characteristic
... class of singularities is closely related to other classes of singularities, the so-called F-singularities, defined using the Frobenius action. For example, locally excellent affine Q-Gorenstein splinters are F-regular by [Sin99], which builds on the Gorenstein case of [HH94]; see also Example 2.4 b ...
... class of singularities is closely related to other classes of singularities, the so-called F-singularities, defined using the Frobenius action. For example, locally excellent affine Q-Gorenstein splinters are F-regular by [Sin99], which builds on the Gorenstein case of [HH94]; see also Example 2.4 b ...
Hilbert`s Nullstellensatz and the Beginning of Algebraic Geometry
... k[X], one can divide one polynoinial f by another polynomial 9 of degree deg 9 = d and get a remainder T such that either T = 0 or deg T < d. The proof given in (ii) of example 2.3 above to show that all ideals in Z are principal (singly generated) works also for k[X], by choosing a polynomial of le ...
... k[X], one can divide one polynoinial f by another polynomial 9 of degree deg 9 = d and get a remainder T such that either T = 0 or deg T < d. The proof given in (ii) of example 2.3 above to show that all ideals in Z are principal (singly generated) works also for k[X], by choosing a polynomial of le ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.