\together with CnoFile {813 325 842 739}{remarks} Let $R$ be an NAD ring (nonassociative and nondistributive ring) and $L$ be the set of all transfinite extensions of $R$ on which some suitable equivalence relations are defined. For any elements $a = (a_0,a_1,\cdots,a_\alpha,\cdots)$, $b =(b_0,b_1,\cdots ,b _\alpha,\cdots )$ of $L$, define $a\oplus b =\break (a_0 + b_0,a_1 + b_1,\cdots,a_\alpha + b _\alpha,\cdots )$ to be the addition and $[a,b]_L$, a subset of $L$, to be the multiplication of $L$. Then after tedious verifications, $L$ becomes an NAD ring. Thus, an NAD ring $L$ is constructed on which the multiplication $f\colon L \times L \to L$ is not necessarily confined to be a single-valued mapping but is rather a general multivalued mapping. The author shows that if $R$ is a Jordan or alternative ring with $[R,R] = R$ then $L$ must be Jordan or alternative. In case $R$ is a general Lie ring with $[R,R]$, then a weak NAD Lie ring $L$ over $R$ is constructed such that the product of any two elements of $L$ is a subset of $L$ and moreover $R$ itself is a proper subgroup of $L$. \par \{Part III has been reviewed [MR0522786 (81m:17008)]. For Parts V and VI see the following two reviews.\}