Part B2: Examples (pp4-8)
... a ring which has the property of being the initial object in the category of rings. In other words, for any ring R there is a unique ring homomorphism φ : Z → R. ...
... a ring which has the property of being the initial object in the category of rings. In other words, for any ring R there is a unique ring homomorphism φ : Z → R. ...
3.3 Factor Rings
... Lemma 3.3.2 shows that the different cosets of I in R are disjoint subsets of R. We note that their union is all of R since every element a of R belongs to some coset of I in R : a ∈ a+ I. The set of cosets of I in R is denoted R/I. We can define addition and multiplication in R/I as follows. Let a ...
... Lemma 3.3.2 shows that the different cosets of I in R are disjoint subsets of R. We note that their union is all of R since every element a of R belongs to some coset of I in R : a ∈ a+ I. The set of cosets of I in R is denoted R/I. We can define addition and multiplication in R/I as follows. Let a ...
Whole Numbers Extending The Natural Numbers Integer Number
... OVERPAID. • If line 54 is more than line 61, subtract line 61 from line 54. This is the [positive] amount you OWE. ...
... OVERPAID. • If line 54 is more than line 61, subtract line 61 from line 54. This is the [positive] amount you OWE. ...
Rings of Fractions
... Theorem 49. Let R be a commutative ring. Let D be any nonempty subset of R that does not contain 0, does not contain any zero divisors, and is closed under multiplication. Then there exists a commutative ring Q with 1 such that Q contains R as a subring and every element of D is a unit in Q. Theorem ...
... Theorem 49. Let R be a commutative ring. Let D be any nonempty subset of R that does not contain 0, does not contain any zero divisors, and is closed under multiplication. Then there exists a commutative ring Q with 1 such that Q contains R as a subring and every element of D is a unit in Q. Theorem ...
MS Word
... 21}. It’s easy to see that since every element is a multiple of 3 (i.e. 3x or 3y, where x, y are elements of the set {0, 1, …7}) adding any two elements will also result in a multiple of 3. ex: 6 = 3*2; 15 = 3*5. 3*2 + 3*5 = 3(5+2) = 21 = 3*7. So closure holds. Similarly, the group identity (0) is p ...
... 21}. It’s easy to see that since every element is a multiple of 3 (i.e. 3x or 3y, where x, y are elements of the set {0, 1, …7}) adding any two elements will also result in a multiple of 3. ex: 6 = 3*2; 15 = 3*5. 3*2 + 3*5 = 3(5+2) = 21 = 3*7. So closure holds. Similarly, the group identity (0) is p ...
Chapter 2 Introduction to Finite Field
... Definition (Field). A field is a ring R in which 1 6= 0 and every non-zero element is invertible. Theorem. Zm is a field ⇐⇒ m is a prime number. Definition 2.1 (Finite field and Order of finite field). A finite field is a field F which has a finite number of elements, this number being called the or ...
... Definition (Field). A field is a ring R in which 1 6= 0 and every non-zero element is invertible. Theorem. Zm is a field ⇐⇒ m is a prime number. Definition 2.1 (Finite field and Order of finite field). A finite field is a field F which has a finite number of elements, this number being called the or ...
