  
  [1X1 [33X[0;0YLie [22Xp[122X[101X[1X-rings[133X[101X
  
  [33X[0;0YIn  this  preliminary  chapter we recall some of the theoretic background of
  Lie rings and Lie [22Xp[122X-rings. We refer to Chapter 5 in [Khu98] for some further
  details. Throughout we assume that [22Xp[122X stands for a rational prime.[133X
  
  [33X[0;0YA  Lie  ring  [22XL[122X  is  an additive abelian group with a multiplication that is
  alternating,  bilinear  and  satisfies  the  Jacobi  identity. We denote the
  product of two elements [22Xg[122X and [22Xh[122X of [22XL[122X with [22Xg h[122X.[133X
  
  [33X[0;0YA  subset  [22XI  ⊆  L[122X  is an [13Xideal[113X in the Lie ring [22XL[122X if it is a subgroup of the
  additive group of [22XL[122X and it satisfies [22Xa l ∈ I[122X for all [22Xa ∈ I[122X and [22Xl ∈ L[122X. As the
  multiplication  in  [22XL[122X  is alternating, it follows that [22Xl a ∈ I[122X for all [22Xl ∈ L[122X
  and  [22Xa ∈ I[122X. Note that if [22XI[122X and [22XJ[122X are ideals in [22XL[122X, then [22XI + J = { a + b ∣ a ∈
  I, b ∈ J}[122X and [22XI J = ⟨ a b ∣ a ∈ I, b ∈ J ⟩_+[122X are ideals in [22XL[122X.[133X
  
  [33X[0;0YA  subset  [22XU  ⊆  L[122X  is  a  [13Xsubring[113X of the Lie ring [22XL[122X if [22XU[122X is a Lie ring with
  respect  to  the  addition  and the multiplication of [22XL[122X. Every ideal in [22XL[122X is
  also  a subring of [22XL[122X. As usual, for an ideal [22XI[122X in [22XL[122X the quotient [22XL/I[122X has the
  structure of a Lie ring, but this does not hold for subrings.[133X
  
  [33X[0;0YThe  [13Xlower  central  series[113X  of  the  Lie ring [22XL[122X is the series of ideals [22XL =
  γ_1(L)  ≥  γ_2(L)  ≥  ...[122X  defined  by [22Xγ_i(L) = γ_i-1(L) L[122X. We say that [22XL[122X is
  [13Xnilpotent[113X  if  there  exists  a  natural  number  [22Xc[122X with [22Xγ_c+1(L) = {0}[122X. The
  smallest natural number with this property is the [13Xclass[113X of [22XL[122X.[133X
  
  [33X[0;0YThe  notion  of  nilpotence now allows us to state the central definition of
  this  package.  A  [13XLie  [22Xp[122X-ring[113X  is  a Lie ring that is nilpotent and has [22Xp^n[122X
  elements for some natural number [22Xn[122X.[133X
  
  [33X[0;0YEvery  finite  dimensional  Lie  algebra  over a field with [22Xp[122X elements is an
  example of a Lie ring with [22Xp^n[122X elements. Note that there exist non-nilpotent
  Lie  algebras of this type: the Lie algebra consisting of all [22Xn × n[122X matrices
  with  trace  [22X0[122X  and  [22Xn  ≥  3[122X is an example. Thus not every Lie ring with [22Xp^n[122X
  elements  is  nilpotent.  (In  contrast to the group case, where every group
  with [22Xp^n[122X elements is nilpotent!)[133X
  
  [33X[0;0YFor  a  Lie  [22Xp[122X-ring  [22XL[122X  we  define  the series [22XL = λ_1(L) ≥ λ_2(L) ≥ ...[122X via
  [22Xλ_i+1(L)  = λ_i(L) L + p λ_i(L)[122X. This series is the [13Xlower exponent-[22Xp[122X central
  series[113X  of [22XL[122X. Its length is the [13X[22Xp[122X-class[113X of [22XL[122X. If [22X|L/λ_2(L)| = p^d[122X, then [22Xd[122X is
  the  [13Xminimal  generator  number[113X  of  [22XL[122X. Similar to the [22Xp[122X-group case, one can
  observe  that this is indeed the cardinality of a generating set of smallest
  possible size.[133X
  
  [33X[0;0YEach  Lie  [22Xp[122X-ring  [22XL[122X  has  a  central  series [22XL = L_1 ≥ ... ≥ L_n ≥ {0}[122X with
  quotients  of  order  [22Xp[122X.  Choose [22Xl_i ∈ L_i ∖ L_i+1[122X for [22X1 ≤ i ≤ n[122X. Then [22X(l_1,
  ..., l_n)[122X is a generating set of [22XL[122X satisfying that [22Xp l_i ∈ L_i+1[122X and [22Xl_i l_j
  ∈  L_i+1[122X for [22X1 ≤ j < i ≤ n[122X. We call such a generating sequence a [13Xbasis[113X for [22XL[122X
  and we say that [22XL[122X has [13Xdimension[113X [22Xn[122X.[133X
  
