Maths cafe

 Def. Ring with unity :  A ring R ¹ {0} is said to be a ring with unity if it contains multiplicative identity i.e., there exists an element denoted by 1Î R such that a. 1= a ==  1.a  for all a  ÎR . Def. Division ring or Skew-field : A ring R is said to be a division ring or skew-field if (i)                   R is with unity i.e. 1 Î R . (ii)                 Every non-zero element of R has a multiplicative inverse i.e., for every non-zero element a   Î R there exists an element b   Î Rsuch that a. b= b. a   = 1. We usually denote multiplicative inverse of a by a-1

    Commutative ring   :  A ring in which a. b =b.a for all a, b     Î R is called a commutative ring i.e., a ring is said to be commutative if it is commutative w.r.t. multiplication.

Def.  Ring with zero divisors :  A ring which contains zero divisors is called a ring with zero divisors. Def .  Ring without zero divisors :  A ring in which no zero divisor exists is called a Ring  OR A ring is called without zero divisors if product of any two non - zero elements is always non- zero.

 Def.  Field :   commutative division ring is called a field. In other words, a commutative ring with unity in which every non-zero element has a multiplicative inverse is called a field.

 

 

Normal subgroups:  Every group 𝐺 has at least two normal subgroups namely the identity {𝑒} and the group 𝐺 itself. The  normal subgroups which are different from these two subgroups are called proper normal subgroups.  All the subgroups of an abelian group are normal. The non-abelian groups all of whose subgroups are  normal are called Hamiltonian Groups. Examplese a) The group 𝑄 = {±1, ±𝑖, ±𝑗, ±𝑘} of quaternions is such that it is non-abelian but every subgroup of  𝑄 is normal. b) The centre of any group is normal. Since 𝜁(𝐺) = {𝑎 ∈ 𝐺 ∶ 𝑎𝑔 = 𝑔𝑎, ∀ 𝑔 ∈ 𝐺}, therefore 𝑔𝜁(𝐺)𝑔−1 = {𝑎 ∈ 𝐺 ∶ 𝑔𝑎𝑔−1 = 𝑎, ∀ 𝑔 ∈ 𝐺}.