Rings of Size 4

Rings with additive group Z₄

• Ring 4.U.1

  Factor ring: Z/4Z={0,1,2,3}
  • Unity
  • Commutative
  • One pair of zero divisors: (2)(2)=0
  • No nontrivial idempotents

• Ring 4.NU.1

  Matrix ring: With coefficients from Z₄,

  0 0 0 0

  ,

  a= 0 0 1 0

  ,

  0 0 2 0

  ,

  0 0 3 0

  • Generated by a; a²=0 ("trivial multiplication")
  • Commutative
  • No unity
  • Every pair is a zero-divisor pair
  • No nontrivial idempotents

• Ring 4.NU.2

  As a subring of Z₈: {0,2,4,6}
  Matrix ring: With coefficients from Z₄,

  0 0 0 0

  ,

  a= 1 1 1 1

  ,

  2 2 2 2

  ,

  3 3 3 3

  • Generated by a; a²=2a
  • Commutative
  • No unity
  • Five pairs of zero divisors
  • No nontrivial idempotents

Commutative rings with additive group Z₂+Z₂, no unit

• Ring 22.NU.1

  Matrix presentation: With coefficients from Z₄,

  0 0 0 0

  ,

  2 0 0 0

  ,

  0 0 0 2

  ,

  2 0 0 2

  • No unity
  • Trivial multiplication
  • No nontrivial idempotents

• Ring 22.NU.2

  As a subring of Z₂+Z₄: {(0,0), a=(0,2), b=(1,0), (1,2)}
  • No unity
  • Five pairs of zero-divisors (a²=ab=ba=a(a+b)=(a+b)a=0)
  • One nontrivial idempotent (b²=b)

• Ring 22.NU.3

  As a subring of Z₄[x]/<2x,x²+x>: {0,2,x,2+x}
  • No unity
  • Five pairs of zero-divisors
  • No nontrivial idempotents

Commutative rings with additive group Z₂+Z₂, with unit

• Ring 22.U.1

  Ring direct sum: Z₂+Z₂
  Factor ring: Z₂[x]/<x²+x>
  Matrix presentation: With coefficients from Z₂,

  0 0 0 0

  ,

  1 0 0 1

  ,

  1 0 0 0

  ,

  0 0 0 1

  • Unity
  • Two pairs of zero divisors
  • Two nontrivial idempotents

• Ring 22.U.2

  Factor Ring: Z₂[x]/<x²+1>={0,1,x,1+x}
  Matrix Ring: With coefficients from Z₂,

  0 0 0 0

  ,

  1 0 0 1

  ,

  0 1 1 0

  ,

  1 1 1 1

  • Unity
  • One pair of zero divisors: (x+1)(x+1)=0
  • No nontrivial idempotents

• Ring 22.U.3

  Factor ring: Z₂[x]/<x²+x+1>={0,1,x,1+x}
  Matrix ring: With coefficients from Z₂,

  0 0 0 0

  ,

  1= 1 0 0 1

  ,

  x= 1 1 1 0

  ,

  0 1 1 1

  • Unity
  • No zero divisors (it's a field)
  • No nontrivial idempotents

Non-commutative rings with additive group Z₂+Z₂

• Ring 22.NC.1

  
  Matrix ring: With coefficients from Z₂,

  0 0 0 0

  ,

  a= 1 0 0 0

  ,

  b= 0 1 0 0

  ,

  1 1 0 0

  • No multiplicative identity
  • Non-commutative (ab = b but ba = 0)
  • Two left-identities (a and a+b) but no right-identities.
  Another representation as a matrix ring (coefficients in Z₂):

  0 0 0 0

  ,

  a= 1 0 1 0

  ,

  b= 1 1 1 1

  ,

  0 1 0 1

  A final representation:

  0 0 0 0

  ,

  a= 0 0 0 1

  ,

  b= 0 0 1 0

  ,

  0 0 1 1

• Ring 22.NC.2

  
  Matrix ring: With coefficients from Z₂,

  0 0 0 0

  ,

  a= 1 0 0 0

  ,

  b= 0 0 1 0

  ,

  1 0 1 0

  • No multiplicative identity
  • Non-commutative (ab = 0 but ba = b)
  • Two right-identities (a and a+b) but no left-identities.
  Another representation as a matrix ring (coefficients in Z₂):

  0 0 0 0

  ,

  a= 0 0 1 1

  ,

  b= 1 1 1 1

  ,

  1 1 0 0

  A final representation:

  0 0 0 0

  ,

  a= 0 0 0 1

  ,

  b= 0 1 0 0

  ,

  0 1 0 1