On invertible matrices over a near-field
In this note it is shown that the inverse of a bijective $2\times 2$ matrix over a specific near-field is not necessarily a matrix. The approach followed here corrects an erroneous argument that was used in [\textit{W.-F. Ke} et al., Proc. R. Soc. Edinb., Sect. A, Math. 140, No. 1, 83--99 (2010; Zbl 1211.16040)] to show the same result. This is attempt to provide an answer to a question -- in case of a matrix near-ring whether a subring of a ring is closed under inverses. In case of a near-ring $R$ and for a positive integer $k$, $R^k$ denotes the direct sum of $k$ copies of the additive group of $R$. Its typical element is of the type with $r_i\in R$ for $1\leq i\leq k$. Moreover, the $i$th coordinate injection function and projection function are respectively as follows: \(\iota_i:R\to R^k\) and \(\pi_i:R^k \to R\). Here $r$ appears in the $i$-th co-ordinate and zeroes elsewhere. For a near-ring $R$ with identity, and a positive integer $k$, $k\times k$ matrix near-ring is the subnear-ring of $M(R^k)$ generated by the set of elementary matrices defined by with $1\leq i$, $j\leq k$ and, for $v\in R^k$. Following a method by \textit{J. D. P. Meldrum} and \textit{A. P. J. van der Walt} [Arch. Math. 47, 312--319 (1986; Zbl 0611.16025)], near-rings of matrix maps are defined for general near-rings, not necessarily with identity. The influence of one-sided identities is discussed. When the base near-ring is integral and planar, the near-ring of matrix maps is shown to be simple. Various types of primitivity of the near-ring of matrix maps are discussed when the base near-ring is planar but not integral. Finally, an open problem concerning bijective matrix maps is solved. An argument regarding the bounddedness on the degree of a certain set of elements from $R$ is used to show that the inverse map $U^{-1}$ does not belong to $M_2(R)$. However it was shown by the first author that, this argument does not hold in a specific case. The result attempted here follows through two lemmas (A and B). For proof the main result the notion of weight of a matrix is introduced. The proof takes its way through this notion of induction on weights. As a consequence to the above result, the authors obtain Corollary 2.4: the inverse map $U^{-1}\in M(R^2)$ does not belong to $M_2(R)$
scientific article; zbMATH DE number 7040796