I'm reading Algebra I by Gorodentsev and I'm having trouble untangling the relationships between alternation, symmetry and skew-symmetry. I will not show all of my work below for brevity's sake.
Let $M$ be an arbitrary module over commutative ring with unity $R$.
Definition: Multilinear map $\omega \colon \prod_{i = 1}^n M \to R$ is alternating if
\begin{align*} \omega(m_1, \dots, m_i, \dots, m_j, \dots, m_n) = -\omega(m_1, \dots, m_j, \dots, m_i, \dots, m_n). \end{align*}
Definition: Multilinear map $\omega \colon \prod_{i = 1}^n M \to R$ is skew-symmetric if it's zero whenever at least two of its arguments coincide.
Definition: Multilinear map $\omega \colon \prod_{i = 1}^n M \to R$ is symmetric if it's invariant under permutations of its arguments.
Claims relating skew-symmetry and alternation: Skew-symmetry always implies alternation. Alternation implies skew-symmetry when the characteristic of $R$ is $2$, and $1_R + 1_R$ is not a zero divisor of $R$. Are these claims correct?
Claims relating symmetry and alternation: When the characteristic of $R$ is $2$, symmetry and alternation are the same.
I am unsure if this latter claim is true, and if so, how to show it. I'll show my work for this below. Consider alternating, multilinear map $\omega \colon \prod_{i = 1}^n M \to R$. Then
\begin{align*} \omega(m_1, \dots, m_i, \dots, m_j, \dots, m_n) &= -\omega(m_1, \dots, m_j, \dots, m_i, \dots, m_n)\\ \implies \omega(m_1, \dots, m_i, \dots, m_j, \dots, m_n) + \omega(m_1, \dots, m_j, \dots, m_i, \dots, m_n) &= 0_R. \end{align*}
I'm unsure of what to do at this point. I appreciate any help.
The terminology for skew-symmetric and alternating you describe in that book looks backwards to me. For $n \geq 2$, I'd say an $n$-multilinear mapping is alternating when it vanishes on all $n$-tuples having an equal pair of coordinates and skew-symmetric when it changes sign after the values in any two different coordinate positions of an $n$-tuple are swapped.
With this in mind, I'd express your first claim as "alternating implies skew-symmetry". For the other direction, keeping in mind the meaning I use for the terms, I'd say "skew-symmetry implies alternating when $2$ is a unit", not when $2 = 0$ ("characteristic is $2$").
See Definition 2.1 and Theorems 2.7, 2.8, and 2.10 here. (The proof of Theorem 2.10 shows the condition that $2$ is a unit can be weakened to (edit) $2n = 0 \Rightarrow n = 0$ in the module $N$, and there is a comment about that in the paragraph right after the proof.)
Are you sure you correctly wrote the definitions of both alternating and skew symmetry from Gorodentsev's book? For example, one of the axioms for a Lie algebra over the real or complex numbers is often expressed as $[y,x] = -[x,y]$, but for Lie algebras over fields of characteristic $2$ this condition is useless and the correct definition of a Lie algebra over a general field should have $[y,x] = -[x,y]$ (skew symmetry) replaced by $[x,x] = 0$ (alternating) and this is called an alternating condition on the Wikipedia page on Lie algebras.