My Page

• hello
  • how are you?
    • level 3
    • level 4
      • level 5 \(\frac{a}{b}\)
  • how are you 1?
• hello 2
  • welcome

helllo

• Krishna Groups
  • 1 Algebraic Structure 2
  • 2 Group Definition 2
  • 3 Abelian Group
  • 4 Finite and infinite group 3
  • 5 General properties of a group
  • 6 Definitino of group based on left axioms 12
  • 7 Composition tables for finite sets.
  • 8 Addition modulo m 33
  • 9 Multiplication modulo p 34
  • 10 Residue classes of set of integers 41
  • 11 An alternative set of postulates for a group 47
  • 12 Permutations
  • 13 Groups of permutations 54
  • 14 Cyclic permutations 56
  • 15 Even and odd permutations 60
  • 16 Integral powers of an element of a group 74
  • 17 Order of an element of a group 75
  • 18 Homomorphism and Isomorphism of groups 92
  • 19 Properties of Homo and Isomorphis 100
  • 20 Relation of Isomorphism in set of all groups 108
  • 21 Transference of group structure 109
  • 22 Complexes and subgroups of a group 113
  • 23 Algebra of complexes
  • 24 Criteria for complexes to be subgroups 116
    • THM1 2 step method; closed + inverse
    • THM2 1 step method; H is subgroup if a,b \(\in\) H => a-b \(\in\) H
  • 25 Criteria for product of subgroups to be subgroup 120
  • 26 Intersection of subgroups 120
  • 27 Cosets 130
    • Properties 2 and 6
  • 28 Relation of congruence modulo 136
  • 29 Lagrange's Theorem 138
  • 30 Order of the product of two subgroups of finite order 141
  • 31 Cayleys Theorem 151
  • 32 Cyclic Groups 153
• Krishna Rings
  • 1. Rings 174
  • 1. Elementary properties of rings 175
  • 1. Integral multiples of elements of rings 176
  • 1. Examples of rings 179
  • 1. Some special types of rings 179
  • 1. Integral domain 181
    2. 42 examples
  • 1. Isomorphism of rngs 216
  • 1. properties of isomorphism of rings 218
  • 1. Subrings 219
    2. If subset itelf is a ring then it is subring
    3. THM The necessary and sufficient conditno for non empty subset S of R to be subring for a,b \(\in\) S we have (a) a-b \(\in\) S and (b) ab \(\in\) S
    4. ie. it is closed under subtration and multiplication
  • 1. Subfields 224
  • 1. Characteristics of ring 228
  • 1. Ordered Integral Domains 231
  • 1. Ideals 236
    2. 21 egs
  • 1. More about ideals 251
  • 1. Ideals generated by a given subset of a ring 252
  • 1. Principal Ideal ring 254
    2. a comm ring without zero divisors and with unity is prinicpal ideal ring if every ideal S in R is principal ideal ie all S = (a)
  • 1. Divisibility in an integral domain 255
    2. Units = are those elements of ring which possess multiplicative inverse
  • 1. Polynomial rings 259
    2. 3 egs
  • 1. Degree of the sum and product of two polynomials 263
  • 1. Rings of polynomials 264
  • 1. Polynomials over integral domain 265
  • 1. Polynomials over a field 266
    2. 3 egs
• Krishna Rings Continued
  • 1. Divisibility of polynomials over a field
  • 1. Division algo for polynomials over a field
    2. 1 eg
  • 1. Euclidean algo for polynomials over a field
  • 1. Unique factorization domain 6
  • 1. Unique factorization thm for polynomials over a field 7
  • 1. Value of polynomial at x = c
  • 1. Quotient Rings or Rings of Residue Classes 17
  • 1. Homo of rings 19
  • 1. Kernel of Ring homo 20
    2. 4 egs
  • 1. Maximal Ideal 24
  • 1. Some more results on ideals 25
  • 1. Prime Ideals 28
    2. Let R be a ring and S an ideal in R ; then S is said to be a prime ideal if ab \(\in\) S , where a,b \(\in\) R => a \(\in\)S or b \(\in\) S
    3. R/S is ID iff S is prime ideal
    4. R is comm ring with unity => maximal ideals are prime ideals
  • 1. Euclidean Rings and Euclidean Domain 33
  • 1. Properties of Euclidean Rings 36
• Krishna Normal Subgroups
  • 1. Normal Subgroups
  • 1. Conjugate elments 63
  • 1. Conjugate subgroups 70
  • 1. Quotient groups 71
  • 1. Homo of groups 77
  • 1. Kernel of homo 80
  • 1. More reuls on group homo 87