Morphisms of Algebraic Stacks

... (1) f is separated, (2) ∆f is a closed immersion, (3) ∆f is proper, or (4) ∆f is universally closed. Proof. The statements “f is separated”, “∆f is a closed immersion”, “∆f is universally closed”, and “∆f is proper” refer to the notions defined in Properties of Stacks, Section 3. Choose a scheme V a ...

... (1) f is separated, (2) ∆f is a closed immersion, (3) ∆f is proper, or (4) ∆f is universally closed. Proof. The statements “f is separated”, “∆f is a closed immersion”, “∆f is universally closed”, and “∆f is proper” refer to the notions defined in Properties of Stacks, Section 3. Choose a scheme V a ...

Oka and Ako Ideal Families in Commutative Rings

... The goal of the present paper is to study more systematically the hierarchical relationships between “Oka”, “Ako”, their strong analogues, and some other properties (Pi ) introduced in [LR: (2.7)]. These properties (Pi ) (i = 1, 2, 3) are recalled in §2, where (P2 ) and (P3 ) are given more streamli ...

... The goal of the present paper is to study more systematically the hierarchical relationships between “Oka”, “Ako”, their strong analogues, and some other properties (Pi ) introduced in [LR: (2.7)]. These properties (Pi ) (i = 1, 2, 3) are recalled in §2, where (P2 ) and (P3 ) are given more streamli ...

Intro Abstract Algebra

... The intersection of two sets A; B is the collection of all elements which lie in both sets, and is denoted A \ B . Two sets are disjoint if their intersection is . If the intersection is not empty, then we may say that the two sets meet. The union of two sets A; B is the collection of all elements ...

... The intersection of two sets A; B is the collection of all elements which lie in both sets, and is denoted A \ B . Two sets are disjoint if their intersection is . If the intersection is not empty, then we may say that the two sets meet. The union of two sets A; B is the collection of all elements ...

Commutative Algebra I

... By a ring R, we mean a (nonempty) set with two binary operations (addition and multiplication) satisfying the following conditions: (1) (R, +) is an abelian group, (2) multiplication is associative, i.e., for all elements x, y, and z in R, x(yz) = (xy)z, and distributive over addition, i.e., for all ...

... By a ring R, we mean a (nonempty) set with two binary operations (addition and multiplication) satisfying the following conditions: (1) (R, +) is an abelian group, (2) multiplication is associative, i.e., for all elements x, y, and z in R, x(yz) = (xy)z, and distributive over addition, i.e., for all ...

abstract algebra: a study guide for beginners - IME-USP

... In this section, it is important to remember that although working with congruences is almost like working with equations, it is not exactly the same. What things are the same? You can add or subtract the same integer on both sides of a congruence, and you can multiply both sides of a congruence by ...

... In this section, it is important to remember that although working with congruences is almost like working with equations, it is not exactly the same. What things are the same? You can add or subtract the same integer on both sides of a congruence, and you can multiply both sides of a congruence by ...

Abstract Algebra - UCLA Department of Mathematics

... extra property. That is, a field has inverses under multiplication: if x is in the field and isn’t 0, then there must be an element x−1 = 1/x as well, and it satisfies x · x−1 = 1. In particular, Q, R, and C are fields as well as rings, but Z is not a field. In a field, fractions add and multiply in ...

... extra property. That is, a field has inverses under multiplication: if x is in the field and isn’t 0, then there must be an element x−1 = 1/x as well, and it satisfies x · x−1 = 1. In particular, Q, R, and C are fields as well as rings, but Z is not a field. In a field, fractions add and multiply in ...