What is ascending chain condition in commutative algebra?

In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.

What does commutative function mean?

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed.

What does commutative mean in addition?

The commutative property of addition says that changing the order of addends does not change the sum.

What is a chain abstract algebra?

In algebraic topology, a k -chain is a formal linear combination of the k -cells in a cell complex. In simplicial complexes (respectively, cubical complexes), k -chains are combinations of k -simplices (respectively, k -cubes), but not necessarily connected.

What is a faithful module?

Faithful. A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.

What do you mean by commutative and distributive?

Since multiplication is commutative, you can use the distributive property regardless of the order of the factors. The Distributive Properties. For any real numbers a, b, and c: Multiplication distributes over addition: a(b + c) = ab + ac. Multiplication distributes over subtraction: a(b – c) = ab – ac.

What is commutative fraction?

The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example a+b=b+a \text{ and\,} (a)(b)=(b)(a). fraction.

What does commutative mean in math?

: being a property of a mathematical operation (as addition or multiplication) in which the result does not depend on the order of the elements The commutative property of addition states that 1 + 2 and 2 + 1 will both have a sum of 3. Which is the correct spelling? Can you spell these 10 commonly misspelled words?

What is the commutative property of addition?

What is commutative property? Give examples. In Mathematics, a commutative property states that if the position of integers are moved around or interchanged while performing addition or multiplication operations, then the answer remains the same. What is commutative property of addition?

What is the Distributive and non-commutative property of multiplication?

The distributive property of Multiplication states that multiplying a sum by a number is same as multiplying every addend by the value and adding the products then. According to the Distributive Property, if a, b, c are real numbers: a x (b + c) = (a x b) + (a x c) Non-Commutative Property. Some operations are non-commutative.

What is the meaning of ascending order?

Illustrated definition of Ascending Order: Arranged from smallest to largest. Increasing. Example: 3, 9, 12, 55 are in ascending…