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Lecture19 - Lecture 19 6 Abstract algebra and coding...

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Lecture 19 6 Abstract algebra and coding theory (continued) 1 Groups 2 Rings and fields 1/14
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Groups XXVIII We sometimes use the following terminology: Definition A left coset g = gH in a group G for a subgroup H is also called a class . An element x g is said to represent the class. In particular we have the following: Definition A class x = x + n Z in Z is called a rest class modulo n . If two classes x and y are equal we say that x is congruent to y modulo n . This is equivalent to x y being divisible by n and denoted by x y mod n . The group Z / n Z is called the group of rest classes modulo n . 2/14
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Rings and fields I Recall that a group is a set G with one operation that satisfies certain properties. We will now define a class of objects with two operations. Definition A ring is a set R with two operations + : R × R R , ( a , b ) mapsto→ a + b · : R × R R , ( a , b ) mapsto→ a · b , called addition and multiplication and two elements 0 R and 1 R such that the following conditions are satisfied: 1 ( R , + , 0) is an abelian group. 2 ( R , · , 1) is a monoid. 3 The distributive laws are satisfied, i.e. ( a + b ) · c = a · c + b · c c · ( a + b ) = c · a + c · b , for all a , b , c R . 3/14
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Rings and fields II Definition A ring is called commutative if multiplication is commutative. A ring is called a skew field if ( R \ { 0 } , · , 1) forms a group, i.e. every element except 0 has a multiplicative inverse. A skew field is called a field if multiplication is commutative. Hence a field is a set R with two operations such that 1 ( R , + , 0) is an abelian group. 2 ( R , · , 1) is a monoid and ( R \ { 0 } , · , 1) is an abelian group. 3 The distributive laws are satisfied. We will usually only consider rings which are commutative. Sometimes one considers rings which do not have a 1. However, all rings we consider have a 1. If R is a ring we denote by R the set of elements which have a multiplicative inverse. Hence in a skew field R = R \ { 0 } .
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