Integral domains in which nonzero locally principal ideals are
... [13], stated below for integral domains (with our addition of the last statement), every semi-quasi-local domain is an LPI domain. In fact, in a semi-quasi-local domain a locally principal ideal is actually principal [20, Theorem 60]. However, the result that we would really like, namely that a loca ...
... [13], stated below for integral domains (with our addition of the last statement), every semi-quasi-local domain is an LPI domain. In fact, in a semi-quasi-local domain a locally principal ideal is actually principal [20, Theorem 60]. However, the result that we would really like, namely that a loca ...
Algebraic Number Theory, a Computational Approach
... Finitely generated abelian groups arise all over algebraic number theory. For example, they will appear in this book as class groups, unit groups, and the underlying additive groups of rings of integers, and as Mordell-Weil groups of elliptic curves. In this section, we prove the structure theorem f ...
... Finitely generated abelian groups arise all over algebraic number theory. For example, they will appear in this book as class groups, unit groups, and the underlying additive groups of rings of integers, and as Mordell-Weil groups of elliptic curves. In this section, we prove the structure theorem f ...
Rings and modules
... Denote the set of all A -bilinear maps f : M × N → R by BilA (M, N ; R). The latter is an A -module with respect to the sum of maps and multiplication of a map by an element of A. Similarly one can define A - n -linear maps. Example. Let A = F be a field, and let M be an F -vector space of dimension ...
... Denote the set of all A -bilinear maps f : M × N → R by BilA (M, N ; R). The latter is an A -module with respect to the sum of maps and multiplication of a map by an element of A. Similarly one can define A - n -linear maps. Example. Let A = F be a field, and let M be an F -vector space of dimension ...
Algebra II (MA249) Lecture Notes Contents
... • Additive groups, where we replace ◦ by +, we denote the identity element by 0, and we denote the inverse of g by −g. If there is more than one group around, and we need to distinguish between the identity elements of G and H say, then we will denote them by 1G and 1H (or 0G and 0H ). Additive grou ...
... • Additive groups, where we replace ◦ by +, we denote the identity element by 0, and we denote the inverse of g by −g. If there is more than one group around, and we need to distinguish between the identity elements of G and H say, then we will denote them by 1G and 1H (or 0G and 0H ). Additive grou ...
Constructive Complete Distributivity II
... of doing Mathematics in a topos and in this more general context the very denition above and Theorem 1 suggest that ( ) is the relevant notion. Indeed, in any topos, P is ( ) for all while the statement \P is ( ) for all " is equivalent to choice. The power objects, P , in a topos are not, in gener ...
... of doing Mathematics in a topos and in this more general context the very denition above and Theorem 1 suggest that ( ) is the relevant notion. Indeed, in any topos, P is ( ) for all while the statement \P is ( ) for all " is equivalent to choice. The power objects, P , in a topos are not, in gener ...
essay
... Theorem 1.2 (Szemerédi’s Theorem). Let Λ ⊆ Z be a set of positive upper Banach density. Then Λ contains arithmetic progressions of arbitrary length. Szemerédi’s proof of this theorem is long and difficult and we will not concern ourselves with it here. Instead, we investigate an alternative, more ...
... Theorem 1.2 (Szemerédi’s Theorem). Let Λ ⊆ Z be a set of positive upper Banach density. Then Λ contains arithmetic progressions of arbitrary length. Szemerédi’s proof of this theorem is long and difficult and we will not concern ourselves with it here. Instead, we investigate an alternative, more ...
Homological Conjectures and lim Cohen
... The direct summand conjecture implies the canonical element conjecture [41], a very strong form of the new intersection which we refer to here as the strong intersection theorem (it has also been called the improved new intersection theorem), which is discussed in the next section, and many other re ...
... The direct summand conjecture implies the canonical element conjecture [41], a very strong form of the new intersection which we refer to here as the strong intersection theorem (it has also been called the improved new intersection theorem), which is discussed in the next section, and many other re ...
NATURAL EXAMPLES OF VALDIVIA COMPACT SPACES 1
... linearly ordered Valdivia compact spaces are hereditarily Valdivia, which is a great difference from general Valdivia compacta. In Section 4 we collect some results on compact groups. They are closely related to Valdivia compact spaces by the theorem of Ivanovskii and Kuźmine saying that compact gr ...
... linearly ordered Valdivia compact spaces are hereditarily Valdivia, which is a great difference from general Valdivia compacta. In Section 4 we collect some results on compact groups. They are closely related to Valdivia compact spaces by the theorem of Ivanovskii and Kuźmine saying that compact gr ...
Connectedness and local connectedness of topological groups and
... All spaces under consideration are assumed to be Hausdorff if no separation axioms are mentioned. If X is a space, then T (X) is its topology and T (x, X) = {U ∈ T (X) : x ∈ U }. The cardinals are identified with the relevant ordinals and are therefore, the sets of all preceding ordinals. In particu ...
... All spaces under consideration are assumed to be Hausdorff if no separation axioms are mentioned. If X is a space, then T (X) is its topology and T (x, X) = {U ∈ T (X) : x ∈ U }. The cardinals are identified with the relevant ordinals and are therefore, the sets of all preceding ordinals. In particu ...
Algebra
... It is important to observe that there are modules which are not free. For example, let A = k[x] and let V be a finite-dimensional k-vector space. Then, as we have seen above, any k-linear map x b : V ! V gives V the structure of a k[x]-module. If the space V has finite dimension over k then the resu ...
... It is important to observe that there are modules which are not free. For example, let A = k[x] and let V be a finite-dimensional k-vector space. Then, as we have seen above, any k-linear map x b : V ! V gives V the structure of a k[x]-module. If the space V has finite dimension over k then the resu ...
Families of elliptic curves of high rank with nontrivial torsion group
... The cases (1.1.6)–(1.1.10) were obtained from the equalities [4]P = [−3]P , [4]P = [−4]P , [5]P = [−4]P , [5]P = [−5]P , [5]P = [−6]P , respectively, which give curves of genus 0 in b and c, hence parametrisable. We suggest a different parametrisation for the cases Z/9Z, Z/10Z and Z/12Z, which will ...
... The cases (1.1.6)–(1.1.10) were obtained from the equalities [4]P = [−3]P , [4]P = [−4]P , [5]P = [−4]P , [5]P = [−5]P , [5]P = [−6]P , respectively, which give curves of genus 0 in b and c, hence parametrisable. We suggest a different parametrisation for the cases Z/9Z, Z/10Z and Z/12Z, which will ...