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Ring {Algebra} (B.Sc. Mathematics NEP Notes)

  Ring RING     A  ring is algebraic structure (R, '+' , '.') that consists of a set equipped with two binary operations, usually called addition and multiplication. The definition of a ring includes the following properties: Additive Closure:  The set R is closed under addition, meaning that if you add any two elements from the set R, the result is also in the set R. Additive Associativity:  Addition is an associative, meaning that for any elements a, b, and c in the set R s.t. (a+b)+c=a+(b+c) Additive Inverse:  For every element  a  in the set R, there exists an additive inverse (usually denoted as  -a ) in the set R such that a+(-a)=0 and (-a)+a=0. Additive Identity:  There exists an additive identity element (usually denoted as 0) in the set R, such that for any element  a  in the set R, a+0=a and 0+a=a. Commutative:  The commutative law for addition states that for any elements a and b in a set R s.t....
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Normal Subgroup {Algebra} (B.Sc. Mathematics NEP Notes)

 NNormal Subgroup Normal Subgroup      Let G be a group, and H be a subgroup of G. Then subgroup H is set to be normal subgroup of G if for any element g in & h in H s.t.  ghg−1 Є H   Simple Group      A simple group is a type of group in abstract algebra that has no non-trivial normal subgroups. In other words, a group G is simple if the only normal subgroups of G are the trivial subgroup {e} and the whole group G itself.   Hamiltonian Group A non-abelian group in which each sub-Group is normal. Conjugate Element      In group theory, given a group G and elements a , b in G the element b is said to be conjugate to "a" if there exists an element g in G such that, b= gag −1 The process of finding such a g is called conjugation. Normalizer of an element      The normalizer of an element  a  in a group G, denoted as N(a), is defined as the set of all elements in G that, which commutes wit...
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Group {Algebra} (B.Sc. Mathematics NEP Notes)

Group Semi group A semigroup is an algebraic structure consisting of a non-empty set equipped with a binary operation (an operation that takes two elements from the set and produces another element in the set) that is associative. In other words, for all elements a, b, and c in the set, the operation satisfies the associative property: (a * b) * c = a * (b * c) In a semigroup, there is no requirement for an identity element or inverses, as is the case in groups. Quasi group Let G be any non-empty set and * is a binary operation than the structure is called Groupoid or quasi group if the binary operation in the set G satisfy the commutative property. a*b = b*a A groupoid with identity element is called Loop. Monoid A monoid is an algebraic structure that consists of a non-empty set, a binary operation A monoid is defined by the following Properties: Associativity:  The binary operation is associative, meaning that for all elements a, b, and c in the set M, the...
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