yes
pure mathematics is mathematics studied and used reasons other than application. So some algebra certainly might be, but not all algebra. Modern or abstract algebra would fit better in that category.
Linear algebra concerns vector spaces whether finite- or infinite-dimensional. Abstract algebra, or modern algebra, includes linear algebra, along with many other kinds of objects, such as groups, rings, fields, lattices, and so on. In part, it was an attempt to put mathematics on a more rigorous footing. Please see the links.
S5 is indecomposable
albert einstein
Abstract algebra is a field of mathematics that studies groups, fields and rings, which all belong to algebraic structures. Algebraic structure and abstract algebra are actually close to each other due to their similarity in topics.
Edward M'William Patterson has written: 'Elementary abstract algebra' -- subject(s): Algebra 'Elementary abstract algebra [by] E.M. Patterson [and] D.E. Rutherford' -- subject(s): Abstract Algebra, Algebra, Abstract
yes
John A. Beachy has written: 'Abstract algebra' -- subject(s): Abstract Algebra, Algebra, Abstract 'Introductory lectures on rings and modules' -- subject(s): Modules (Algebra), Noncommutative rings
Gertrude Ehrlich has written: 'Fundamental concepts of abstract algebra' -- subject(s): MATHEMATICS / Algebra / Abstract, Abstract Algebra 'Fundamental concepts of abstract algebra' -- subject(s): Abstract Algebra 'Fundamental concepts of abstract algebra' -- subject(s): MATHEMATICS / Algebra / Abstract, Abstract Algebra
Dennis Kletzing has written: 'Abstract algebra' -- subject(s): Abstract Algebra
George Mackiw has written: 'Applications of abstract algebra' -- subject(s): Abstract Algebra
Gary D. Crown has written: 'Abstract algebra' -- subject(s): Abstract Algebra
John W. Keesee has written: 'Elementary abstract algebra' -- subject(s): Abstract Algebra
yes, also this question belongs in the linear algebra forum not the abstract algebra forum
D. S. Malik has written: 'Java Programming' 'Java programming' -- subject(s): Java (Computer program language) 'Fundamentals of abstract algebra' -- subject(s): Abstract Algebra, Algebra, Abstract 'C++ Programming'
David Ronald Dickinson has written: 'Operators: an algebraic synthesis' -- subject(s): Abstract Algebra, Algebra, Abstract