[Like most Russian mathematicians there are different ways to transliterate Malcev's name into the Roman alphabet. The most common way, other than Malcev, is to write it as Maltsev.]
Anatoly Ivanovich Malcev's father was a glass-blower so his background was certainly from that of a poor family. It did not take long for his mathematical abilities to shine, however, and his teachers at secondary school quickly became convinced that he was destined to become an outstanding mathematician.
Malcev graduated from school in 1927 and, in the same year, he went to Moscow State University to study mathematics. He graduated in 1931 but before that he had already begun to teach in a secondary school in Moscow in 1930. After graduating he continued with his teaching career then, in 1932, he was appointed as an assistant at the Ivanovo Pedagogical Institute which was in Ivanovo to the north-east of Moscow. The town, on both banks of the Uvod River, had been known as Ivanovo-Voznesensk but was renamed Ivanovo in 1932, the year Malcev started to work there.
When Malcev began teaching in Ivanovo he had only a first degree but Ivanovo had a good rail link with Moscow and he was able to make frequent trips there to discuss his research with Kolmogorov. Malcev's first publications were on logic and model theory and resulted from work he had begun entirely on his own. Ideas from these papers were later to reappear in Robinson's work on non-standard analysis.
Kolmogorov invited Malcev to join his graduate programme at Moscow University, and he held a studentship there for a year although he continued his teaching post at the Ivanovo Pedagogical Institute during the year. Malcev always considered himself to have been Kolmogorov's student and certainly during the year he held the studentship at Moscow University, Malcev was directed by Kolmogorov to certain algebra problems.
In 1937 Malcev published a paper on the embeddability of a ring in a field, answering a question posed by Kolmogorov. The question had been posed originally by van der Waerden as to whether there existed rings without zero-divisors which could not be embedded in a field. Malcev answered this question by constructing a ring whose multiplicative semigroup was not embeddable in a group. He was led to investigate the existence of rings whose multiplicative semigroup was embeddable in a group yet the ring still was not embeddable in a field. This led, two years later, to another fundamental paper of Malcev where he gave necessary and sufficient conditions for a semigroup to be embeddable in a group.
In 1937 Malcev wrote a dissertation on Torsion free abelian groups of finite rank then, between 1939 and 1941, he studied for his doctorate (which in Russia is similar to the habilitation) at the Steklov Institute of the USSR Academy of Sciences. During this period in which he undertook doctoral research Malcev continued as a lecturer at the Ivanovo Pedagogical Institute. In 1941 he received the degree of Doctor of Science for a dissertation Structure of isomorphic representable infinite algebras and groups.
Malcev became a professor at Ivanovo Pedagogical Institute in 1944. His work in group theory continued and he proved important results on linear groups and, in particular, on linear soluble groups. In  and  his work on representations of infinite groups by matrices is commented on:-
Among these results we mention the local theorem for the class of groups representable by matrices of a given order, and also the theorem on residual finiteness of finitely generated linear groups. This last theorem implies, in particular, the proposition that free groups are residually finite.
Malcev also studied Lie groups and topological algebras, producing a synthesis of algebra and mathematical logic. For example he wrote on semisimple subgroups of Lie groups in 1944 and Free topological algebras in 1957. In 1946 he was awarded a State Prize for his work on Lie groups. Malcev also created a synthesis of the theory of algebras and of algorithms called constructive algebras.
In 1960, Malcev was appointed to a chair in mathematics at the Mathematics Institute at Novosibirsk and to be chairman of the Algebra and Logic Department at Novosibirsk State University. At Novosibirsk ( and ):-
Malcev founded the Siberian section of the Mathematics Institute of the Academy of Sciences, a logic-algebraic school with many members, and directed the world famous seminar Algebra i Logika. In 1962 Malcev founded the specialised journal Algebra i Logika and from that time he was editor of this journal. He was also the founder of the Siberian Mathematical Society and its first president, chief editor and active leader of Sibirsk. Mat. Zh.; and a member of one of the oldest journals in our country, Mat. Sbornik.
During the early 1960s Malcev worked on problems of decidability of elementary theories of various algebraic structures. He showed the undecidability of the elementary theory of finite groups, of free nilpotent groups, of free soluble groups and many others. He investigated the undecidability of the elementary theory of classical linear groups and proved that the class of locally free algebras had a decidable theory.
In 1948 Malcev wrote the undergraduate text Foundation of linear algebra which appeared in English translation in 1962. If I [EFR] may add a personal note, when I began to teach linear algebra in around 1970, I discovered that the publisher Freeman gave away free copies of texts that lecturers might consider adopting for use by a class. I received Malcev's book Foundation of linear algebra from them and I was astonished to find this remarkable approach to linear algebra in a book superbly written to give an understanding of the subject rather than the approach adopted by a large number of linear algebra books which seem on the whole written to provide good examination questions.
At the regrettably early age of 57 Malcev died while taking part in the Novosibirsk Topology Conference which he had helped to organise ( and ):-
It is impossible to forget the sorrow and deep concern that Malcev's death caused to all the mathematicians, those who lived in Novosibirsk as well as the guests. All felt they were saying farewell to a great scientist, to whom many were indebted...
Just before his death Malcev had delivered his final lecture at this Novosibirsk Topology Conference. It was on algebras, now called Malcev algebras, which are natural generalisations of Lie algebras. Malcev had introduced these algebras in 1955 and in this lecture he gave a survey of his work on this topic over the twelve preceding years.
Malcev received many honours, and world-wide recognition for his innovative work. He was elected a member of the USSR Academy of Sciences in 1958. We mentioned some of the prizes he received above, such as the State Prize in 1946, but another important honour which he received in 1964 was a Lenin Prize for his series of papers on the applications of mathematical logic to algebra.
In  and  Malcev's contribution is summed up by pointing out that his:-
... mathematical work is distinguished by the abundance of new ideas and the creation of new mathematical trends on the one hand, and the solution of a number of classical problems on the other hand.
He is described as:-
... a person of great charm, an interesting companion and a wise counsellor on scientific and worldly affairs.
In  some of Malcev's interests are described:-
Malcev liked very long walks and long distance swimming. He played the violin from a very early age until he was 30, when he switched to the piano ... He liked history, and there are a number of volumes on this subject in his library, including the history of mathematics.
Other interests that he had included poetry and his particular tastes in music included Bach and Russian folk songs.
Article by: J J O'Connor and E F Robertson
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