Tables of finite commutative local rings of small orders

of small orders

Andrzej Nowicki 08.08.2018, tab-05

All rings are assumed to be commutative with unity. It is well known that every finite ring is a finite product of finite local rings, and the orders of finite local rings are powers of primes. We present some descriptions, up to isomorphism, of finite local rings of orders pⁿ, for p ∈ {2, 3, 5, 7} and n ∈ {1, 2, 3, 4, 5}. Sources: papers [1] - [13].

Local rings of order 2

|R| = 2¹ A₁ = Z2

|R| = 2² B₁ = F4

B₂ = Z2[x]/(x²) B₃ = Z4

|R| = 2³ C₁ = F8

C2 = Z²[x]/(x³)

C₃ = Z2[x, y]/(x², xy, y²) C₄ = Z4[x]/(2x, x²) C5 = Z⁴[x]/(2x, x²− 2) C₆ = Z8

There are, up to isomorphism, exactly 21 commutative local rings R of order 16.

|R| = 2⁴, char (R) = 2 D₁ = F16,

D2 = F⁴[x]/(x²), D₃ = Z2[x]/(x⁴), D₄ = Z2[x, y]/(x², y²), D₅ = Z2[x, y]/(x², y²− xy), D₆ = Z2[x, y]/(x³, xy, y²), D₇ = Z2[x, y, z]/(x, y, z)².

|R| = 2⁴, char (R) = 2³ D₁₉= Z8[x]/(2x, x²), D₂₀= Z8[x]/(2x, x²+ 4).

|R| = 2⁴, char (R) = 2⁴ D₂₁= Z16.

|R| = 2⁴, char (R) = 2² D₈ = Z4[x]/(x²),

D₉ = Z4[x]/(x²+ 2), D₁₀ = Z4[x]/(x²+ 1), D11 = Z⁴[x]/(x²+ 3), D₁₂ = Z4[x]/(x²+ x + 1), D₁₃ = Z4[x]/(x³, 2x), D₁₄ = Z4[x]/(x³− 2, 2x),

D₁₅ = Z4[x, y]/(x², y², xy, 2x, 2y), D₁₆ = Z4[x, y]/(x²− 2, y², xy, 2x, 2y), D17 = Z⁴[x, y]/(x², y², xy − 2, 2x, 2y), D₁₈ = Z4[x, y]/(x²− 2, y²− 2, xy, 2x, 2y).

There are, up to isomorphism, exactly 55 commutative local rings R of order 32.

|R| = 2⁵, char (R) = 2¹ E₁ = F32,

E₂ = Z2[x]/(x⁵),

E₃ = Z2[x, y]/(x⁴, xy, y²− x³), E4 = Z²[x, y]/(x⁴, xy, y²), E₅ = Z2[x, y, z, t]/(x, y, z, t)².

E₆ = Z2[x, y]/(x³, x²y, y²), E₇ = Z2[x, y]/(x³, x²y, y²− xy), E₈ = Z2[x, y]/(x³, x²y, y²− x²− xy), E9 = Z²[x, y, z]/(x³, y², z², xy, xz, yz), E₁₀ = Z2[x, y, z]/(x³, y²− x², z², xy, xz, yz), E₁₁ = Z2[x, y, z]/(x³, y²− x², z²− x², xy, xz, yz), E₁₂ = Z2[x, y, z]/(x³, y², z², xy, yz).

|R| = 2⁵, char (R) = 2² E₁₃= Z4[x]/(2x, x⁴− 2), E₁₄= Z4[x]/(2x², x³− 2),

E₁₅= Z4[x, y]/(x³, y², xy + 2, 2x, 2y), E₁₆= Z4[x, y]/(x³, x²+ y², xy + 2, 2x, 2y), E17= Z⁴[x, y]/(x³, x²+ y²+ 2, xy + 2, 2x, 2y), E₁₈= Z4[x, y]/(x³, y² + 2, xy, 2x, 2y),

E₁₉= Z4[x, y]/(xy², x², y²+ 2, xy, 2x, 2y), E₂₀= Z4[x, y]/(x³, x²+ xy, y²+ 2, xy, 2x, 2y), E₂₁= Z4[x, y]/(x³− 2, y², xy, 2x, 2y)

E₂₂= Z4[x, y]/(x³− 2, y²− 2, xy, 2x, 2y), E23= Z⁴[x, y]/(x²− 2, y², xy, 2y);

E₂₄= Z4[x, y]/(x²− 2, y²− 2x, xy, 2y), E₂₅= Z4[x, y]/(x²− 2x − 2, y², xy, 2y),

E₂₆= Z4[x]/(x⁴, 2x), E₂₇= Z4[x]/(x³, 2x²), E₂₈= Z4[x]/(x³− 2x, 2x²),

E₂₉= Z4[x, y, z]/(x²− 2, y², z², xy, xz, yz, 2x, 2y, 2z), E₃₀= Z4[x, y, z]/(x²− 2, y²− 2, z², xy, xz, yz, 2x, 2y, 2z), E₃₁= Z4[x, y, z]/(x²− 2, y²− 2, z²− 2, xy, xz, yz, 2x, 2y, 2z), E₃₂= Z4[x, y, z]/(x², y², z², xy, xz, yz − 2, 2x, 2y, 2z),

E₃₃= Z4[x, y, z]/(x²− z, y², z², xy, xz − 2, yz, 2x, 2y, 2z), E₃₄= Z4[x, y, z]/(x²− z, y²− 2, z², xy, xz − 2, yz, 2x, 2y, 2z), E₃₅= Z4[x, y, z]/(x², y², z², xy − z, xz, yz, 2x, 2y, 2z),

E₃₆= Z4[x, y]/(x², y², xy, 2y), E₃₇= Z4[x, y]/(x², y²− 2x, xy, 2y), E₃₈= Z4[x, y]/(x²− 2x, y², xy, 2y), E₃₉= Z4[x, y, z]/(2, x, y, z)².

|R| = 2⁵, char (R) = 2³ E₄₀= Z8[x]/(2x, x³), E41= Z⁸[x]/(4x, x²− 2), E₄₂= Z8[x]/(4x, x²− 2x − 2), E₄₃= Z8[x]/(4x, x²− 2x + 2), E₄₄= Z8[x]/(2x, x³− 4), E₄₅= Z8[x]/(2x, x³),

E46= Z⁸[x]/(4x, x²), E₄₇= Z8[x]/(4x, x²− 2x), E₄₈= Z8[x]/(4x, x²− 2x − 4), E₄₉= Z8[x, y]/(x², y², xy, 2x, 2y), E₅₀= Z8[x, y]/(x²− 4, y², xy, 2x, 2y), E₅₁= Z8[x, y]/(x²− 4, y²− 4, xy, 2x, 2y), E52= Z⁸[x, y]/(x², y², xy − 4, 2x, 2y).

|R| = 2⁵, char (R) = 2⁴ E₅₃= Z16[x]/(2x, x²), E₅₄= Z16[x]/(2x, x²− 8).

|R| = 2⁵, char (R) = 2² E₅₅ = Z32.

Local rings of order 3

|R| = 3¹ A₁ = Z3

|R| = 3² B₁ = F9

B₂ = Z3[x]/(x²) B₃ = Z9

|R| = 3³ C₁ = F27

C₂ = Z3[x]/(x³)

C₃ = Z3[x, y]/(x², xy, y²) C₄ = Z9[x]/(3x, x²) C₅ = Z9[x]/(3x, x²− 3) C₆ = Z9[x]/(3x, x²− 6) C7 = Z²⁷

There are, up to isomorphism, exactly 22 commutative local rings R of order 3⁴.

|R| = 3⁴, char (R) = 3 D₁ = F81,

D2 = F⁹[x]/(x²), D₃ = Z3[x]/(x⁴),

D₄ = Z3[x, y]/(x²− y², xy), D₅ = Z3[x, y]/(x²− 2y², xy), D₆ = Z3[x, y]/(x³, xy, y²), D₇ = Z3[x, y, z]/(x, y, z)².

|R| = 3⁴, char (R) = 3² D₈ = Z9[x]/(x²),

D9 = Z⁹[x]/(x²− 2), D₁₀= Z9[x]/(x²− 3), D₁₁= Z9[x]/(x²− 6), D₁₂= Z9[x]/(3x, x³), D13= Z⁹[x]/(3x, x³− 3),

D₁₄= Z9[x, y]/(x², y², xy, 3x, 3y), D₁₅= Z9[x, y]/(x²− 3, y², xy, 3x, 3y), D₁₆= Z9[x, y]/(x²− 6, y², xy, 3x, 3y), D₁₇= Z9[x, y]/(x²− 3, y²− 3, xy, 3x, 3y), D₁₈= Z9[x, y]/(x²− 3, y²− 6, xy, 3x, 3y).

|R| = 3⁴, char (R) = 3³ D₁₉ = Z27[x]/(3x, x²), D₂₀ = Z27[x]/(3x, x²− 9), D21 = Z²⁷[x]/(3x, x²− 18).

|R| = 3⁴, char (R) = 3⁴ D₂₂= Z81.

There are, up to isomorphism, exactly 66 commutative local rings R of order 3⁵.

|R| = 3⁵, char (R) = 3¹ E₁ = F3⁵,

E2 = Z³[x]/(x⁵),

E₃ = Z3[x, y]/(x⁴, xy, y²− x³), E₄ = Z3[x, y]/(x⁴, xy, y²), E₅ = Z3[x, y, z, t]/(x, y, z, t)²,

E₆ = Z3[x, y]/(x³, y², x²y), E₇ = Z3[x, y]/(x³, x²− y², x²y), E₈ = Z3[x, y]/(x³, x²− 2y², x²y),

E₉ = Z3[x, y, z]/(x³, y², z², xy, xz, yz), E₁₀= Z3[x, y, z]/(x³, x²− y², z², xy, xz, yz), E11= Z³[x, y, z]/(x³, x²− 2y², z², xy, xz, yz), E₁₂= Z3[x, y, z]/(x³, y² − x², z²− x², xy, xz, yz).

|R| = 3⁵, char (R) = 3² E₁₃ = Z9[x]/(3x, x⁴− 3), E₁₄ = Z9[x]/(3x, x⁴− 6),

E₁₅= Z9[x]/(3x², x³− 3), E₁₆= Z9[x]/(3x², x³+ 3x − 3), E17= Z⁹[x]/(3x², x³− 3x − 3), E₁₈ = Z9[x, y]/(x³, xy, y²+ 3, 3x, 3y),

E₁₉ = Z9[x, y]/(x³, xy, y²+ 6, 3x, 3y), E₂₀ = Z9[x, y]/(x³, xy, y²− x²+ 3, 3x, 3y), E₂₁ = Z9[x, y]/(x³, xy, y²− 2x²+ 3, 3x, 3y), E₂₂ = Z9[x, y]/(x³, xy, y²− 2x²+ 6, 3x, 3y), E₂₃ = Z9[x, y]/(x³, xy + 3, y², 3x, 3y),

E24 = Z⁹[x, y]/(x³, xy + 3, y²− x²+ 3, 3x, 3y), E₂₅ = Z9[x, y]/(x³, xy + 3, y²− 2x², 3x, 3y), E₂₆ = Z9[x, y]/(x³, xy + 3, y²− 2x²+ 3, 3x, 3y), E₂₇ = Z9[x, y]/(x³, xy + 3, y²− 2x²+ 6, 3x, 3y).

E₂₈= Z9[x, y, z]/(x²− z, y², z², xy, xz − 3, yz, 3x, 3y, 3z), E₂₉= Z9[x, y, z]/(x²− z, y²− 3, z², xy, xz − 3, yz, 3x, 3y, 3z), E₃₀= Z9[x, y, z]/(x²− z, y²− 6, z², xy, xz − 3, yz, 3x, 3y, 3z), E₃₁ = Z9[x, y]/(x³− 3, y², xy, 3y);

E₃₂ = Z9[x, y]/(x²− 6, y², xy, 3y);

E₃₃ = Z9[x, y]/(x²− 3, y² − 3x, xy, 3y), E₃₄ = Z9[x, y]/(x²− 6, y² − 3x, xy, 3y).

E₃₅ = Z9[x]/(x⁴, 3x), E₃₆ = Z9[x]/(x³, 3x²), E₃₇ = Z9[x]/(x³− 3x, 3x²), E₃₈ = Z9[x]/(x³− 6x, 3x²), E₃₉= Z9[x, y, z]/(x²− 3, y², z², zy, xz, yz, 3x, 3y, 3z),

E₄₀= Z9[x, y, z]/(x²− 6, y², z², xy, xz, yz, 3x, 3y, 3z);

E₄₁= Z9[x, y, z]/(x²− 3, y²− 3, z², zy, xz, yz, 3x, 3y, 3z), E₄₂= Z9[x, y, z]/(x²− 3, y²− 6, z², xy, xz, yz, 3x, 3y, 3z), E₄₃= Z9[x, y, z]/(x²− 3, y²− 3, z²− 3, zy, xz, yz, 3x, 3y, 3z), E44= Z⁹[x, y, z]/(x²− 3, y²− 3, z²− 6, xy, xz, yz, 3x, 3y, 3z),

E₄₅= Z9[x, y, z]/(x²− z, y², z², xy, xz, yz, 3x, 3y, 3z), E₄₆= Z9[x, y, z]/(x²− z, y²− z, z², xy, xz, yz, 3x, 3y, 3z), E47= Z⁹[x, y, z]/(x²− z, y²− 2z, z², xy, xz, yz, 3x, 3y, 3z),

E₄₈= Z9[x, y]/(x², y², xy, 3y), E₄₉= Z9[x, y]/(x², y²− 3x, xy, 3y), E₅₀= Z9[x, y, z]/(3, x, y, z)².

|R| = 3⁵, char (R) = 3³ E₅₁= Z3³[x]/(3²x, x²− 3), E₅₂= Z3³[x]/(3²x, x²− 6), E₅₃= Z3³[x]/(3x, x³− 3²), E₅₄= Z3³[x]/(3x, x³),

E₅₅ = Z3³[x]/(3²x, x²), E56 = Z3³[x]/(3²x, x²− 3²), E57 = Z3³[x]/(3²x, x²− 2 · 3²),

|R| = 3⁵, char (R) = 3⁴

E₅₈= Z3³[x, y]/(x², y², xy, 3x, 3y), E₅₉= Z3³[x, y]/(x² − 3², y², xy, 3x, 3y), E₆₀= Z3³[x, y]/(x² − 2 · 3², y², xy, 3x, 3y), E61= Z3³[x, y]/(x² − 3², y²− 3², xy, 3x, 3y), E₆₂= Z3³[x, y]/(x² − 3², y²− 2 · 3², xy, 3x, 3y),

E₆₃= Z3⁴[x]/(3x, x²), E₆₄= Z3⁴[x]/(3x, x²− 3³), E₆₅= Z3⁴[x]/(3x, x²− 2 · 3³).

|R| = 3⁵, char (R) = 3⁵ E₆₆= Z3⁵.

Local rings of order 5

|R| = 5¹ A₁ = Z5

|R| = 5² B₁ = F5²

B₂ = Z5[x]/(x²) B3 = Z5²

|R| = 5³ C₁ = F5³

C2 = Z⁵[x]/(x³)

C₃ = Z5[x, y]/(x², xy, y²) C₄ = Z5²[x]/(5x, x²) C5 = Z5²[x]/(5x, x²− 5) C₆ = Z5²[x]/(5x, x²− 10) C₇ = Z5³

There are, up to isomorphism, exactly 22 commutative local rings R of order 5⁴.

|R| = 5⁴, char (R) = 5 D₁ = F5⁴,

D₂ = F5²[x]/(x²), D₃ = Z5[x]/(x⁴),

D₄ = Z5[x, y, z]/(x, y, z)², D₅ = Z5[x, y]/(x³, y², xy), D₆ = Z5[x, y]/(x²− y², xy), D7 = Z⁵[x, y]/(x²− 2y², xy).

|R| = 5⁴, char (R) = 5² D₈ = Z5²[x]/(x²),

D₉ = Z5²[x]/(x²− 2), D₁₀= Z5²[x]/(x²− 5), D₁₁= Z5²[x]/(x²− 10), D12= Z5²[x]/(5x, x³), D₁₃= Z5²[x]/(5x, x³− 5),

D₁₄= Z5²[x, y]/(x², y², xy, 5x, 5y), D₁₅= Z5²[x, y]/(x²− 5, y², xy, 5x, 5y), D16= Z5²[x, y]/(x²− 10, y², xy, 5x, 5y), D₁₇= Z5²[x, y]/(x²− 5, y²− 5, xy, 5x, 5y), D₁₈= Z5²[x, y]/(x²− 5, y²− 10, xy, 5x, 5y),

|R| = 5⁴, char (R) = 5³ D₁₉ = Z5³[x]/(5x, x²), D20 = Z5³[x]/(5x, x²− 5²), D₂₁ = Z5³[x]/(5x, x²− 2 · 5²).

|R| = 5⁴, char (R) = 5⁴ D₂₂= Z5⁴.

There are, up to isomorphism, exactly 68 commutative local rings R of order 5⁵.

|R| = 5⁵, char (R) = 5¹ E₁ = F5⁵,

E₂ = Z5[x]/(x⁵,

E₃ = Z5[x, y]/(x⁴, xy, y²− x³), E₄ = Z5[x, y]/(x⁴, xy, y²), E₅ = Z5[x, y, z, t]/(x, y, z, t)²,

E₆ = Z5[x, y]/(x³, y², x²y) E₇ = Z5[x, y]/(x³, x² − y², x²y) E₈ = Z5[x, y]/(x³, x² − 2y², x²y), E₉ = Z5[x, y, z]/(x³, y², z², xy, xz, yz), E₁₀ = Z5[x, y, z]/(x³, x²− y², z², xy, xz, yz), E11 = Z⁵[x, y, z]/(x³, x²− 2y², z², xy, xz, yz), E₁₂ = Z5[x, y, z]/(x³, y²− x², z²− x², xy, xz, yz).

|R| = 5⁵, char (R) = 5² E₁₃= Z5²[x]/(5x, x⁴− 5), E₁₄= Z5²[x]/(5x, x⁴− 10), E₁₅= Z5²[x]/(5x, x⁴− 15), E16= Z5²[x]/(5x, x⁴− 20), E₁₇= Z5²[x]/(5x², x³− 5).

E₁₈= Z5²[x, y]/(x³, xy, y²+ 5, 5x, 5y), E₁₉= Z5²[x, y]/(x³, xy, y²+ 10, 5x, 5y), E20= Z5²[x, y]/(x³, xy, y²− x²+ 5, 5x, 5y), E₂₁= Z5²[x, y]/(x³, xy, y²− x²+ 10, 5x, 5y), E₂₂= Z5²[x, y]/(x³, xy, y²− 2x²+ 5, 5x, 5y), E₂₃= Z5²[x, y]/(x³, xy + 5, y², 5x, 5y),

E₂₄= Z5²[x, y]/(x⁵, xy + 5, y²− x²+ 5, 5x, 5y), E₂₅= Z5²[x, y]/(x³, xy + 5, y²− x²+ 10, 5x, 5y), E₂₆= Z5²[x, y]/(x³, xy + 5, y²− 2x², 5x, 5y), E27= Z5²[x, y]/(x³, xy + 3, y²− 4x²+ 10, 5x, 5y).

E₂₈ = Z5²[x, y, z]/(x²− z, y², z², xy, xz − 5, yz, 5x, 5y, 5z);

E₂₉ = Z5²[x, y, z]/(x²− z, y²− 5, z², xy, xz − 5, yz, 5x, 5y, 5z);

E30 = Z5²[x, y, z]/(x²− z, y²− 10, z², xy, xz − 5, yz, 5x, 5y, 5z);

E₃₁ = Z5²[x, y]/(x²− 5, y², xy, 5y);

E₃₂ = Z5²[x, y]/(x²− 10, y², xy, 5y);

E33 = Z5²[x, y]/(x²− 5, y²− 5x, xy, 5y), E₃₄ = Z5²[x, y]/(x²− 10, y² − 5x, xy, 5y), E₃₅ = Z5²[x, y]/(x²− 15, y² − 5x, xy, 5y), E₃₆ = Z5²[x, y]/(x²− 20, y² − 5x, xy, 5y).

E₃₇ = Z5²[x]/(x⁴, 5x), E38 = Z5²[x]/(x³, 5x²), E₃₉ = Z5²[x]/(x³− 5x, 5x²), E₄₀ = Z5²[x]/(x³− 10x, 5x²).

E₄₁= Z5²[x, y, z]/(x²− 5, y², z², zy, xz, yz, 5x, 5y, 5z);

E₄₂= Z5²[x, y, z]/(x²− 10, y², z², xy, xz, yz, 5x, 5y, 5z);

E₄₃= Z5²[x, y, z]/(x²− 5, y² − 5, z², zy, xz, yz, 5x, 5y, 5z);

E₄₄= Z5²[x, y, z]/(x²− 5, y² − 10, z², xy, xz, yz, 5x, 5y, 5z);

E₄₅= Z5²[x, y, z]/(x²− 5, y² − 5, z²− 5, zy, xz, yz, 5x, 5y, 5z);

E46= Z5²[x, y, z]/(x²− 5, y² − 5, z²− 105, xy, xz, yz, 5x, 5y, 5z);

E₄₇= Z5²[x, y, z]/(x²− z, y², z², xy, xz, yz, 5x, 5y, 5z);

E₄₈= Z5²[x, y, z]/(x²− z, y²− z, z², xy, xz, yz, , 5x, 5y, 5z);

E₄₉= Z5²[x, y, z]/(x²− z, y²− 2z, z², xy, xz, yz, , 5x, 5y, 5z).

E50 = Z5²[x, y]/(x², y², xy, 5y), E₅₁ = Z5²[x, y]/(x², y²− 5x, xy, 5y), E₅₂ = Z5²[x, y, z]/(5, x, y, z)².

|R| = 5⁵, char (R) = 5³ E53 = Z5³[x]/(5²x, x²− 5), E₅₄ = Z5³[x]/(5²x, x²− 10), E₅₅ = Z5³[x]/(5x, x³ − 5²), E₅₆ = Z5³[x]/(5x, x³), E₅₇ = Z5³[x]/(5²x, x²), E₅₈ = Z5³[x]/(5²x, x²− 5²), E₅₉ = Z5³[x]/(5²x, x²− 2 · 5²).

E₆₀ = Z5³[x, y]/(x², y², xy, 5x, 5y), E₆₁ = Z5³[x, y]/(x²− 5², y², xy, 5x, 5y), E62 = Z5³[x, y]/(x²− 2 · 5², y², xy, 5x, 5y), E₆₃ = Z5³[x, y]/(x²− 5², y²− 5², xy, 5x, 5y), E₆₄ = Z5³[x, y]/(x²− 5², y²− 2 · 5², xy, 5x, 5y).

|R| = 5⁵, char (R) = 5⁴ E₆₅ = Z5⁴[x]/(5x, x²), E₆₆ = Z5⁴[x]/(5x, x²− 5³), E₆₇ = Z5⁴[x]/(5x, x²− 2 · 5³).

|R| = 5⁵, char (R) = 5⁵ E₆₈= Z5⁵.

Local rings of order 7

|R| = 7¹ A₁ = Z7

|R| = 7² B1 = F7²

B₂ = Z7[x]/(x²) B₃ = Z7²

|R| = 7³ C₁ = F7³

C₂ = Z7[x]/(x³)

C3 = Z⁷[x, y]/(x², xy, y²) C₄ = Z7²[x]/(7x, x²) C₅ = Z7²[x]/(7x, x²− 7) C6 = Z7²[x]/(7x, x²− 21) C₇ = Z7³

There are, up to isomorphism, exactly 24 commutative local rings R of order 7⁴.

|R| = 7⁴, char (R) = 7 D1 = F7⁴,

D₂ = F7²[x]/(x²), D₃ = Z7[x]/(x⁴),

D₄ = Z7[x, y, z]/(x, y, z)², D₅ = Z7[x, y]/(x³, y², xy), D₆ = Z7[x, y]/(x²− y², xy), D₇ = Z7[x, y]/(x²− 3y², xy).

|R| = 7⁴, char (R) = 7² D₈ = Z7²[x]/(x²),

D₉ = Z7²[x]/(x²− 3), D₁₀ = Z7²[x]/(x²− 7), D₁₁ = Z7²[x]/(x²− 21), D12 = Z7²[x]/(7x, x³), D₁₃ = Z7²[x]/(7x, x³− 7), D₁₄ = Z7²[x]/(7x, x³− 14), D₁₅ = Z7²[x]/(7x, x³− 21),

D₁₆= Z7²[x, y]/(x², y², xy, 7x, 7y), D₁₇= Z7²[x, y]/(x²− 7, y², xy, 7x, 7y), D₁₈= Z7²[x, y]/(x²− 21, y², xy, 7x, 7y), D₁₉= Z7²[x, y]/(x²− 7, y²− 7, xy, 7x, 7y), D₂₀= Z7²[x, y]/(x²− 7, y²− 21, xy, 7x, 7y),

|R| = 7⁴, char (R) = 7³ D₂₁ = Z7³[x]/(7x, x²), D₂₂ = Z7³[x]/(7x, x²− 7²), D23 = Z7³[x]/(7x, x²− 3 · 7²).

|R| = 7⁴, char (R) = 7⁴ D₂₄= Z7⁴.

There are, up to isomorphism, exactly 74 commutative local rings R of order 7⁵.

|R| = 7⁵, char (R) = 7¹ E₁ = F7⁵,

E2 = Z⁷[x]/(x⁵,

E₃ = Z7[x, y]/(x⁴, xy, y²− x³), E₄ = Z7[x, y]/(x⁴, xy, y²), E₅ = Z7[x, y, z, t]/(x, y, z, t)²,

E₆ = Z7[x, y]/(x³, y², x²y) E₇ = Z7[x, y]/(x³, x² − y², x²y) E₈ = Z7[x, y]/(x³, x² − 3y², x²y), E₉ = Z7[x, y, z]/(x³, y², z², xy, xz, yz), E₁₀ = Z7[x, y, z]/(x³, x²− y², z², xy, xz, yz), E₁₁ = Z7[x, y, z]/(x³, x²− 3y², z², xy, xz, yz), E12 = Z⁷[x, y, z]/(x³, y²− x², z²− x², xy, xz, yz).

|R| = 5⁵, char (R) = 5² E13 = Z7²[x]/(7x, x⁴ − 7), E₁₄ = Z7²[x]/(7x, x⁴ − 21), E₁₅ = Z7²[x]/(7x², x³− 7), E₁₆ = Z7²[x]/(7x², x³− 14), E₁₇ = Z7²[x]/(7x², x³− 21).

E18 = Z7²[x, y]/(x³, xy, y²+ 7, 7x, 7y), E₁₉ = Z7²[x, y]/(x³, xy, y²+ 21, 7x, 7y), E₂₀ = Z7²[x, y]/(x³, xy, y²− x²+ 7, 7x, 7y), E₂₁ = Z7²[x, y]/(x³, xy, y²− 3x²+ 7, 7x, 7y), E₂₂ = Z7²[x, y]/(x³, xy, y²− 3x²+ 21, 7x, 7y), E₂₃ = Z7²[x, y]/(x³, xy + 7, y², 7x, 7y),

E₂₄ = Z7²[x, y]/(x³, xy + 7, y²− x²+ 7, 7x, 7y), E25 = Z7²[x, y]/(x³, xy + 7, y²− 3x², 7x, 7y), E₂₆ = Z7²[x, y]/(x³, xy + 7, y²− 3x² + 21, 7x, 7y), E₂₇ = Z7²[x, y]/(x³, xy + 7, y²− 3x² + 28, 7x, 7y).

E28= Z7²[x, y, z]/(x² − z, y², z², xy, xz − 7, yz, 7x, 7y, 7z), E₂₉= Z7²[x, y, z]/(x² − 2z, y², z², xy, xz − 7, yz, 7x, 7y, 7z), E₃₀= Z7²[x, y, z]/(x² − 3z, y², z², xy, xz − 7, yz, 7x, 7y, 7z), E₃₁= Z7²[x, y, z]/(x² − z, y²− 7, z², xy, xz − 7, yz, 7x, 7y, 7z), E₃₂= Z7²[x, y, z]/(x² − 2z, y²− 7, z², xy, xz − 7, yz, 7x, 7y, 7z), E₃₃= Z7²[x, y, z]/(x² − 3z, y²− 7, z², xy, xz − 7, yz, 7x, 7y, 7z), E₃₄= Z7²[x, y, z]/(x² − z, y²− 21, z², xy, xz − 7, yz, 7x, 7y, 7z), E35= Z7²[x, y, z]/(x² − 2z, y²− 21, z², xy, xz − 7, yz, 7x, 7y, 7z), E₃₆= Z7²[x, y, z]/(x² − 3z, y²− 21, z², xy, xz − 7, yz, 7x, 7y, 7z), E37 = Z7²[x, y]/(x²− 7, y², xy, 7y);

E₃₈ = Z7²[x, y]/(x²− 21, y², xy, 7y);

E₃₉ = Z7²[x, y]/(x²− 7, y²− 7x, xy, 7y), E₄₀ = Z7²[x, y]/(x²− 21, y² − 7x, xy, 7y),

E41 = Z7²[x]/(x⁴, 7x), E₄₂ = Z7²[x]/(x³, 7x²), E₄₃ = Z7²[x]/(x³− 7x, 7x²), E₄₄ = Z7²[x]/(x³− 21x, 7x²).

E₄₅ = Z7²[x, y, z]/(x²− 7, y², z², zy, xz, yz, 7x, 7y, 7z), E₄₆ = Z7²[x, y, z]/(x²− 21, y², z², xy, xz, yz, 7x, 7y, 7z), E₄₇ = Z7²[x, y, z]/(x²− 7, y²− 7, z², zy, xz, yz, 7x, 7y, 7z), E₄₈ = Z7²[x, y, z]/(x²− 7, y²− 21, z², xy, xz, yz, 7x, 7y, 7z), E49 = Z7²[x, y, z]/(x²− 7, y²− 7, z² − 7, zy, xz, yz, 7x, 7y, 7z), E₅₀ = Z7²[x, y, z]/(x²− 7, y²− 7, z² − 21, xy, xz, yz, 7x, 7y, 7z),

E₅₁= Z7²[x, y, z]/(x²− z, y², z², xy, xz, yz, 7x, 7y, 7z);

E₅₂= Z7²[x, y, z]/(x²− z, y²− z, z², xy, xz, yz, , 7x, 7y, 7z);

E₅₃= Z7²[x, y, z]/(x²− z, y²− 3z, z², xy, xz, yz, 7x, 7y, 7z).

E₅₄ = Z7²[x, y]/(x², y², xy, 7y), E₅₅ = Z7²[x, y]/(x², y²− 7x, xy, 7y), E₅₆ = Z7²[x, y, z]/(7, x, y, z)².

|R| = 7⁵, char (R) = 7³ E₅₇ = Z7³[x]/(7²x, x²− 7), E₅₈ = Z7³[x]/(7²x, x²− 21), E₅₉ = Z7³[x]/(7x, x³ − 7²), E₆₀ = Z7³[x]/(7x, x³ − 2 · 7²), E61 = Z7³[x]/(7x, x³ − 3 · 7²), E₆₂ = Z7³[x]/(7x, x³),

E₆₃ = Z7³[x]/(7²x, x²), E₆₄ = Z7³[x]/(7²x, x²− 7²), E65 = Z7³[x]/(7²x, x²− 3 · 7²),

E₆₆ = Z7³[x, y]/(x², y², xy, 7x, 7y), E₆₇ = Z7³[x, y]/(x²− 7², y², xy, 7x, 7y), E₆₈ = Z7³[x, y]/(x²− 3 · 7², y², xy, 7x, 7y), E₆₉ = Z7³[x, y]/(x²− 7², y²− 7², xy, 7x, 7y), E₇₀ = Z7³[x, y]/(x²− 7², y²− 3 · 7², xy, 7x, 7y).

|R| = 7⁵, char (R) = 7⁴ E71 = Z7⁴[x]/(7x, x²), E₇₂ = Z7⁴[x]/(7x, x²− 7³), E₇₃ = Z7⁴[x]/(7x, x²− 3 · 7³).

|R| = 7⁵, char (R) = 7⁵ E₇₄= Z7⁵.

References

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Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, 87-100 Toru´n, Poland, (e-mail: anow@mat.uni.torun.pl).