Karteikarten
Rings: Intro and Definitions
Karten in diesem Stapel (25)
- 1A ring isA set with two binary operations called addition and multiplication
- 2The ring axioms are1. The set is an abelian group wrt +
2. The set is closed under *
3. The set is associative under *
4. The two distributive laws of * over + hold - 3A ring with an identity/unity isA ring such that there exists 1
- 4We assume \(0 \neq 1\). True or FalseTrue
- 5Let R be a ring. Let \(a,b,c \in R \). If \(a+b=a+c\) then \(b=c\)There exists \(a^(-1)\) so \(a^(-1)\) added to both sides gives us \(b=c\)
- 60a = 00a + 0a = (0+0)a = 0a = 0a + 0 so adding -0a to both sides, it follows that 0a = 0
- 7(-a)b = -(ab)(-a)b + (ab) = (-a+a)b = 0b = 0 so it means that (-a)b = -(ab)
- 8(-a)(-b)=(ab)(-a)(-b) + -(ab) = (-a)(-b) + (-a)b = (-a)(-b+b) = (-a)0 = 0 so (-a)(-b) = -(-(ab)) = ab
- 9What do you call a ring in which multiplication is commutative?A commutative ring
- 10A ring with unity has a unique multiplicative identitySuppose a and b are both identities. It follows that a = ab = ba = b
- 11{0} is a ring (the trivial ring)0+0=0, 0*0=0
- 12If 0=1 in a ring then the ring is the trivial ringFor all r in R r =1r=0r=0
- 13Let R be a ring. For a \(\neq\)0 in R-{0} if for some b\(\neq\)0 in R, a is a left zero divisor ifab=0
- 14Multiplication by a is left-multiplicative iffa is not a zero divisor
- 15An integral domain isA commutative ring without zero divisors
- 16For a in R, a is a unit ifthere exists a b in R such that ab=ba=1
- 17Prove that U, the set of units, is a group under multiplicationAssume u and v are units
This means there exists r and s in R:
ru=ur=1 and sv=vs=1
a) (sr)(uv)=s(r(uv))=s((ru)v)=s(1v)=sv=1
Similarly (uv)(sr)=1 (uv) is a unit
b) Associativity is given
c) 1*1=1 so 1 is in U
d) \(a^(-1)*a=a*a^(-1)=1\) - 18Prove that a zero divisor is not a unitLet x be a unit and x*y=0, y\(\neq\)0.
\((x^(-1)*x)*y=0\)
\(1*y=0\)
\(y=0\) which is a contradiction - 19A division ring isA ring in which every nonzero element has a multiplicative inverse
- 20Definition of field (in terms of rings)A field is a commutative division ring
- 21Definition of field (in terms of groups)A field is a ring which forms an abelian group under *
- 22If R is an integral domain and we have nonzero elements a,b and c. If ab = ac then b=cTake ab-ac=0 so a(b-c)=0 but since R is an integral domain we do not have zero divisors. So b-c=0 and hence b=c
- 23Every finite integral domain isA field
- 24Subring testa) S is not empty
b) S contains inverses
c) a+b and ab is in S - 25The trivial subring {0} is a subring ofEvery ring
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- Beschreibung
- Undergrad Math Karteikarten am Rings: Intro and Definitions, erstellt von Vincent Tembo am 12/07/2016.
- Erstellt von
- Vincent Tembo
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