Unit of a ring

Unit of a ring
Tag	`019NFGFPR5`
URI	`m2://019NFGFPR5`
Kind	definition
Writing	W1 stated
MSC	16U60
Defines	Unit of a ring
Uses	Ring

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Definition

Let $R$ be a ring. An element $u \in R$ is a unit of $R$ if there is an element $v \in R$ such that $u \cdot v = v \cdot u = 1$ . The element $v$ is unique. It is the inverse of $u$ , with the notation $u^{-1}$ .

Remarks

The units of $\mathbb{Z}$ are $1$ and $-1$ . In a field, every nonzero element is a unit. A real $n \times n$ matrix is a unit of the matrix ring if and only if its determinant is not zero. The units of $R$ form the group of units of $R$ .

Explanations

L1 human

A unit is an element that you can divide by. In the integers, you can divide by $1$ and $-1$ and stay in the integers, but you cannot divide by $2$ . So $1$ and $-1$ are the only units of $\mathbb{Z}$ .

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)

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What this needs

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Group concept
Ring definition
Unit of a ring definition

What depends on this

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

Transcendental element 00SW8B91E7
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
Intermediate field 02NBKVQFRQ
Splitting field 02NZZ1J0FH
Algebraically closed field 02SJG1HC32
Galois field 04SGPNT8R9
Irreducible element 052SDEPZH7
Associate elements 053E86NWVK
PID implies UFD 055HT85T3S
The rationals as the field of fractions of the integers 05K3J8J8T2
Group of units of a ring 05THZKR9AN
Maximal ideals are prime 05WAF5T0GZ
Uniqueness of splitting fields 069A3BC3EP
The minimal polynomial divides every polynomial vanishing at the element 06K7SARZKR
Units of the Gaussian integers 06RH4XAGRY
Unique factorisation domain 06YMXJMXYS
Subfield 07QY5RPFBR
Product of primitive polynomials is primitive 08C1NWBNHA
Finite field 08RTDHX1SK
Content of a polynomial 08VV7YJRB7
Subfields of a finite field 08YERZDM9X
In a PID a nonzero element is irreducible iff it generates a maximal ideal 095ZHQ7WCJ
Algebraic extension 09EYMKMSB9
Ring homomorphisms from a field are injective 09N7DZ7SDB
Tower law 0A7KJ68XHT
K[x]/(f) is a field iff f is irreducible 0AWMPX35CW
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
Finite extensions are algebraic 0DFYTNZ6AM
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
Transcendence of pi 0G0F7FGCGP
The quaternions 0GE4DVQKR8
Minimal polynomial of an algebraic element 0GK1WB9R98
Degree of a field extension 0GS74FGFEK
The order of a finite field is a prime power 0H5WWC8RZX
Galois group of a polynomial 0JZY3R908H
Fixed field 0KMJ6919WS
Reduction modulo p test for irreducibility 0KWWEXW0RN
The only ideals of a field are zero and the whole field 0M6G75E95E
Adjoining a root of a polynomial 0M893X7YY7
Field generated by adjoining elements 0MA5B744K3
Algebraic closure 0MCXX29HWY
Prime element 0MNHXBSX3M
Separable extension 0N019F6RE9
Degree of a simple extension is the degree of the minimal polynomial 0N5QW9K7MH
The minimal polynomial is irreducible 0NFEG48RDR
Algebraic element 0NPW1DWECM
Finite extension 0NRYPM9Y27
Z/pZ is a field for p prime 0NV6E43P4A
Simple extension 0P6SBR7YM2
The constructible numbers form a field closed under square roots 0PN9WMRWT7
Field 0QA8CHH8MG
The algebraic numbers form a field 0QFTETCHKB
Gauss's lemma 0QFV5Y1Z5X
The Galois group has order at most the degree 0QM98DNWVW
In a UFD irreducible elements are prime 0QRCFZKXH8
Every element of a finite field of order q satisfies x^q = x 0QWR8WTZEW
Wedderburn's little theorem 0RRNVX1VKA
Irreducibility of the cyclotomic polynomials over the rationals 0SK1VHZ5R2
Primitive polynomial 0SP14DZN0G
The degree of a constructible number is a power of two 0SSSM1D475
Primitive element theorem 0SVX98HYV1
Fundamental theorem of Galois theory 0T1Q8D0FW8
Existence of finite fields of every prime power order 0T8M3YC5RG
An algebraic extension of an algebraic extension is algebraic 0TPWS0K1BC
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
Frobenius automorphism of a finite field 0VP4TBVJ19
Prime subfield 0VPFBAY7MW
Bézout's identity in a Euclidean domain 0VY3WPP9Y3
R/M is a field iff M is maximal 0WD3SE7S1X
Field extension 0WQ1A8SHTZ
Division ring 0WZXK05ME7
Impossibility of squaring the circle 0X7CVDE3BG
Galois group 0XVR72M51M
Gauss–Wantzel theorem 0Y34JMJNE9
Primitive element of a finite field 0YAA9CJQXF
K(a) is isomorphic to K[x] modulo the minimal polynomial 0ZN39QDTP9

Sources

Web https://en.wikipedia.org/wiki/Unit_(ring_theory), lead section

What links here

Used by

Irreducible element 052SDEPZH7
Associate elements 053E86NWVK
Group of units of a ring 05THZKR9AN
Units of the Gaussian integers 06RH4XAGRY
Prime element 0MNHXBSX3M
Field 0QA8CHH8MG
Division ring 0WZXK05ME7