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25 Cards in this Set
- Front
- Back
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Ring
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a ring R is a nonempty set with 2 binary operations such that
1 a + b = b + a 2 ( a + b ) + c = a + ( b + c) 3 there is an additive identity 0. 4 every element has an additive inverse 5 a(bc) = (ab)c a(b+c)=ab+acand (b+c)a=ba+ca |
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unity; identity
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non zero element that is an identity under multiplication
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unity and inverses are unique
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but they need not exist.
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subring
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a nonempty subset S of a ring R is a subring if S is closed under subtraction and multiplication. That is if a and b are in S then (a-b) and ab are in S
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Integral Domain
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a commutative ring with unity and no zero divisors (=cancellation law holds)
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Zero Divisor
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nonzero element of a commutative R such that there is a nonzero element b and ab=0
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field
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commutative ring with unity in which every nonzero element is a unit
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Characteristic of a Ring
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the least positive integer n such that nx=0 for all x in R.
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Ring
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a ring R is a nonempty set with 2 binary operations such that
1 a + b = b + a 2 ( a + b ) + c = a + ( b + c) 3 there is an additive identity 0. 4 every element has an additive inverse 5 a(bc) = (ab)c a(b+c)=ab+acand (b+c)a=ba+ca |
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unity; identity
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non zero element that is an identity under multiplication
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unity and inverses are unique
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but they need not exist.
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subring
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a nonempty subset S of a ring R is a subring if S is closed under subtraction and multiplication. That is if a and b are in S then (a-b) and ab are in S
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Integral Domain
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a commutative ring with unity and no zero divisors (=cancellation law holds)
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Zero Divisor
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nonzero element of a commutative R such that there is a nonzero element b and ab=0
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field
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commutative ring with unity in which every nonzero element is a unit
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finite integral domains are fields
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`
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Z_p is a field
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`
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Characteristic of a Ring
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the least positive integer n such that nx=0 for all x in R.
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Characteristic of a Ring with Unity
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R has unity 1. If 1 has infinite order under addition then the characteristic of R is 0. If 1 has order n under addition then the characteristic of R is n.
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Characteristic of an Integral Domain
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is 0 or prime
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Ideal
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A subring A of a ring R is called ideal of R if for every r in R and every a in A borth ra and ar are in A
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Ideal Test
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a-b in A when a and b are in A AND
ra and ar are in A whenever a in A and r in R |
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Factor Ring
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R/A = {r + A | r in R}
R?A is a ring iff A is an ideal of R |
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Prime Ideal
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a,b in R ab in I thus either a or b is in I
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Maximal Ideal
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A is an ideal of R if A<=B<=R then A=B or B=R
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