In this preliminary chapter we recall some of the theoretic background of Lie rings and Lie \(p\)-rings. We refer to Chapter 5 in [Khu98] for some further details. Throughout we assume that \(p\) stands for a rational prime.
A Lie ring \(L\) is an additive abelian group with a multiplication that is alternating, bilinear and satisfies the Jacobi identity. We denote the product of two elements \(g\) and \(h\) of \(L\) with \(g h\).
A subset \(I \subseteq L\) is an ideal in the Lie ring \(L\) if it is a subgroup of the additive group of \(L\) and it satisfies \(a l \in I\) for all \(a \in I\) and \(l \in L\). As the multiplication in \(L\) is alternating, it follows that \(l a \in I\) for all \(l \in L\) and \(a \in I\). Note that if \(I\) and \(J\) are ideals in \(L\), then \(I + J = \{ a + b \mid a \in I, b \in J\}\) and \(I J = \langle a b \mid a \in I, b \in J \rangle_+\) are ideals in \(L\).
A subset \(U \subseteq L\) is a subring of the Lie ring \(L\) if \(U\) is a Lie ring with respect to the addition and the multiplication of \(L\). Every ideal in \(L\) is also a subring of \(L\). As usual, for an ideal \(I\) in \(L\) the quotient \(L/I\) has the structure of a Lie ring, but this does not hold for subrings.
The lower central series of the Lie ring \(L\) is the series of ideals \(L = \gamma_1(L) \geq \gamma_2(L) \geq \ldots\) defined by \(\gamma_i(L) = \gamma_{i-1}(L) L\). We say that \(L\) is nilpotent if there exists a natural number \(c\) with \(\gamma_{c+1}(L) = \{0\}\). The smallest natural number with this property is the class of \(L\).
The notion of nilpotence now allows us to state the central definition of this package. A Lie \(p\)-ring is a Lie ring that is nilpotent and has \(p^n\) elements for some natural number \(n\).
Every finite dimensional Lie algebra over a field with \(p\) elements is an example of a Lie ring with \(p^n\) elements. Note that there exist non-nilpotent Lie algebras of this type: the Lie algebra consisting of all \(n \times n\) matrices with trace \(0\) and \(n \geq 3\) is an example. Thus not every Lie ring with \(p^n\) elements is nilpotent. (In contrast to the group case, where every group with \(p^n\) elements is nilpotent!)
For a Lie \(p\)-ring \(L\) we define the series \(L = \lambda_1(L) \geq \lambda_2(L) \geq \ldots\) via \(\lambda_{i+1}(L) = \lambda_i(L) L + p \lambda_i(L)\). This series is the lower exponent-\(p\) central series of \(L\). Its length is the \(p\)-class of \(L\). If \(|L/\lambda_2(L)| = p^d\), then \(d\) is the minimal generator number of \(L\). Similar to the \(p\)-group case, one can observe that this is indeed the cardinality of a generating set of smallest possible size.
Each Lie \(p\)-ring \(L\) has a central series \(L = L_1 \geq \ldots \geq L_n \geq \{0\}\) with quotients of order \(p\). Choose \(l_i \in L_i \setminus L_{i+1}\) for \(1 \leq i \leq n\). Then \((l_1, \ldots, l_n)\) is a generating set of \(L\) satisfying that \(p l_i \in L_{i+1}\) and \(l_i l_j \in L_{i+1}\) for \(1 \leq j < i \leq n\). We call such a generating sequence a basis for \(L\) and we say that \(L\) has dimension \(n\).
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