Zero divisor

Zero divisor
Tag	`0K26YKNBSA`
URI	`m2://0K26YKNBSA`
Kind	definition
Writing	W1 stated
MSC	16U99
Defines	Zero divisor
Uses	Ring

Markdown · JSON · Edit

Definition

Let $R$ be a ring. An element $a \in R$ is a left zero divisor if there is a nonzero $x \in R$ with $a x = 0$ . An element $a \in R$ is a right zero divisor if there is a nonzero $y \in R$ with $y a = 0$ . An element is a zero divisor if it is a left or a right zero divisor.

Remarks

In a commutative ring, left and right zero divisors are the same. In $\mathbb{Z}/6\mathbb{Z}$ , the classes of $2$ and $3$ are zero divisors, because $2 \cdot 3 = 0$ . Some authors do not count $0$ as a zero divisor. With the definition above, $0$ is a zero divisor of every nonzero ring. A unit is never a zero divisor.

Explanations

L1 human

In the integers, a product is zero only when one factor is zero. In other rings this can fail. In the integers modulo $6$ , $2 \cdot 3 = 6 = 0$ , but neither $2$ nor $3$ is zero.

What this node needs for W2

edge instance_of in: one example (0 of 1)
edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 3 nodes in this order. The foundations come first, and the list ends with this node.

Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.

no evidence level

Group concept
Ring definition
Zero divisor definition

What depends on this

72 nodes use this node, directly or through other nodes.

The integers are a Euclidean domain 000JFQGDXG
The characteristic of an integral domain is zero or prime 00758QM6JJ
Irreducible polynomial 00X2A9G9W3
Uniqueness of finite fields of a given order 010PKCQPXP
Existence of splitting fields 01441KDGBY
Uniqueness of the algebraic closure 01VQ7C4FTB
Euclidean algorithm in a Euclidean domain 023VY493K0
Eisenstein's criterion 02BPKD7TMD
Algebraically closed field 02SJG1HC32
Norm on the Gaussian integers 03AN1V4ZFJ
Galois field 04SGPNT8R9
Irreducible element 052SDEPZH7
PID implies UFD 055HT85T3S
The rationals as the field of fractions of the integers 05K3J8J8T2
Maximal ideals are prime 05WAF5T0GZ
Uniqueness of splitting fields 069A3BC3EP
Z[x] is not a principal ideal domain 06E3AG9TM2
Units of the Gaussian integers 06RH4XAGRY
Unique factorisation domain 06YMXJMXYS
Product of primitive polynomials is primitive 08C1NWBNHA
Content of a polynomial 08VV7YJRB7
In a PID a nonzero element is irreducible iff it generates a maximal ideal 095ZHQ7WCJ
Integral domain 09S3PFBEPM
R/P is an integral domain iff P is prime 0AVXBEC05W
K[x]/(f) is a field iff f is irreducible 0AWMPX35CW
R[x] is an integral domain iff R is 0B5Z1EC66J
Order of the Galois group of a Galois extension equals its degree 0BDJBH8B4Z
Z[√-5] is not a UFD 0BSJYCG7NM
Normal extension 0CBRFQ7DVZ
A finite field of order q is the splitting field of x^q - x 0CDAHW576M
Irreducible elements need not be prime 0CHE3J7QMX
Finite subgroups of the multiplicative group of a field are cyclic 0CKGBEJX6B
Impossibility of doubling the cube 0CQ357M0QS
Polynomial ring over a UFD is a UFD 0D6KH8Y59N
K[x] is a Euclidean domain 0DQG39K2NH
Galois extension 0DS2F7GK05
A finite integral domain is a field 0EBE031B8E
The multiplicative group of a finite field is cyclic 0EBZ3G455G
Existence of an algebraic closure 0ETGBAK2CC
Impossibility of trisecting the angle 0EWXPZCSZ0
Prime elements are irreducible in an integral domain 0F0K8DB9TD
Greatest common divisor in an integral domain 0F7KNJ1ZYN
Euclidean domain 0GJ2HRG95T
Cancellation in an integral domain 0GJM24EXP9
The order of a finite field is a prime power 0H5WWC8RZX
Gaussian integers 0K5W69AC0J
A polynomial over a field has at most as many roots as its degree 0KJVQV3E0F
Reduction modulo p test for irreducibility 0KWWEXW0RN
Adjoining a root of a polynomial 0M893X7YY7
Algebraic closure 0MCXX29HWY
Degree of a simple extension is the degree of the minimal polynomial 0N5QW9K7MH
The minimal polynomial is irreducible 0NFEG48RDR
Z/pZ is a field for p prime 0NV6E43P4A
The Gaussian integers are a Euclidean domain 0PN13BB7BT
Gauss's lemma 0QFV5Y1Z5X
Euclidean domains are principal ideal domains 0QKVE1GD9A
In a UFD irreducible elements are prime 0QRCFZKXH8
Principal ideal domain 0RZYDVZ475
Irreducibility of the cyclotomic polynomials over the rationals 0SK1VHZ5R2
Primitive polynomial 0SP14DZN0G
The degree of a constructible number is a power of two 0SSSM1D475
Fundamental theorem of Galois theory 0T1Q8D0FW8
Degree of a product 0T233AZ1GE
Existence of finite fields of every prime power order 0T8M3YC5RG
Field of fractions 0VF11ZTTMD
Every polynomial has a root in some extension field 0VG02M1FEZ
Polynomials of degree two or three are irreducible iff they have no roots 0VNE161QCK
Bézout's identity in a Euclidean domain 0VY3WPP9Y3
Impossibility of squaring the circle 0X7CVDE3BG
Gauss–Wantzel theorem 0Y34JMJNE9
Primitive element of a finite field 0YAA9CJQXF
K(a) is isomorphic to K[x] modulo the minimal polynomial 0ZN39QDTP9