6.6 Rings and fields 6.6.1 Rings Definition 21: A ring is an Abelian group [R, +] with an...
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Transcript of 6.6 Rings and fields 6.6.1 Rings Definition 21: A ring is an Abelian group [R, +] with an...
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6.6 Rings and fields 6.6.1 Rings Definition 21: A ring is an Abelian group [R, +]
with an additional associative binary operation(denoted · such that for all a, b, cR,
(1) a · (b + c) = a · b + a · c, (2) (b + c) · a = b · a + c · a. We write 0R for the identity element of the group
[R, +]. For a R, we write -a for the additive inverse of a. Remark: Observe that the addition operation is
always commutative while the multiplication need not be.
Observe that there need not be inverses for multiplication.
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1. Identity of ring and zero of ring Theorem 6.27: Let [R;+,*] be an unitary
1 ring. Then the following results hold. (1)a*0=0*a=0 for aR
(2)a*(-b)=(-a)*b=-(a*b) for a,bR (3)(-a)*(-b)=a*b for a,bR (4)(-1)*a=-a for aR (5)(-1)*(-1)=1
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1:Identity of ring 0:zero of ring
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[M2,2(Z);+,] is an unitary ring
Zero of ring (0)22,
Identity of ring is
},,,|{)(2,2 Zdcbadc
baZM
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ring ofdivisor -zero
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2. Zero-divistorsDefinition 23: If a0 is an element of a ring R for which there exists b0 such that ab=0(ba=0), then a is called a left(right) zero-divistor in R.Let S={1,2}, is zero element of ring [P(S);,∩]
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6.6.2 Integral domains, division rings and fields
Definition 24: A commutative ring is an integral domain if there are no zero-divisors.
[P(S);,∩] and [M;+,] are not integral domain, [Z;+,] is an integral domain
Theorem 6.28: If R is an integral domain then for any a, b, cR if a0 and ab=ac, then b=c.
Proof: Suppose that R is an integral domain. If ab = ac, then ab - ac=0
Let [R;+;*] be a ring with identity element 1. If 1=0, then for aR, a=a*1=a*0=0. Hence R has only one element, In other words,
If |R|>1, then 10.
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Definition 25: A ring is a division ring if the non-zero elements form a group under multiplication.
If R is a division ring, then |R|2.
Ring R has identity, and any non-zero element exists inverse element under multiplication.
Definition 26: A field is a commutative division ring.
[Z;+,]is a integral domain, but it is not division ring and field
[Q;+,], [R;+,]and[C;+,] are field
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Let [F;+,*] be a algebraic system, and |F| 2,
(1)[F;+]is a Abelian group (2)[F-{0};*] is a Abelian group (3)For a,b,cF, a*(b+c)=(a*b)+(a*c)
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Let .
Then [M22(Q);+,*] is a division ring. But it is not a field
}0,,,,|{)(2,2
dc
baQdcba
dc
baQM
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Theorem 6.29: Any Field is an integral domain
Let [F;+,*] be a field. Then F is a commutative ring.
If a,b, F-{0}, s.t. a*b =0 。 [Z;+,] is an integral domain. But it is
not a field
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Theorem 6.30: A finite integral domain is a field.
integral domain :commutative, no zero-divisor Field: commutative, identity, inverse identity, inverse Let [R;+,*] be a finite integral domain. (1)Need to find 1R such that 1*a =a for
all a R. (2)For each aR-{0}, need to find an
element bR such that a*b = 1. Proof:(1)Let R={a1,a2,an}. For cR, c 0, consider the set
Rc={a1*c, a2*c, ,an*c}R.
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Exercise:P367 7,8,16,17,20 1. Let Z[i] = {a + bi| a, bZ}. (1)Show that Z[i] is a commutative ring
and find its units. Is (2)Is Z[i] a field? Why? 2.Show that Q[i] = {a + bi | a, bQ} is a
field.