We introduce the notions of strongly zero-product (strongly Jordan zero-product) preserving maps on normed algebras. These notions are generalization of the concepts of zero-product and Jordan zero-product preserving maps. Also for a non-zero vector space V and for a non-zero linear functional f on V, we equip V with a multiplication, converting V into an associative algebra, denoted by Vf . We characterize the zero-product (Jordan zero-product) preserving maps on Vf . Also we characterize the strongly zero-product (strongly Jordan zero-product) preserving maps on Vf in the case where V is a normed vector space and f is a continuous linear functional on V. Finally, for polynomials in one variable x over Vf , we shall show that each polynomial of precise degree n ≥ 0, with non-zero constant term has precisely n-zeros (counted with multiplicity) in Vf . While, polynomials of precise degree n ≥ 2, with zero constant term have infinitely many zeros when dim(V) ≥2. This shows that the algebraic fundamental theorem for polynomial equations over an arbitrary algebra, is not valid in general.
Khoddami, A. R. (2015). Strongly Zeo-product Preserving Maps on Normed Algebras Induced by a Bounded Linear Functional. Khayyam Journal of Mathematics, 1(1), 107-114. doi: 10.22034/kjm.2015.12290
MLA
Ali Reza Khoddami. "Strongly Zeo-product Preserving Maps on Normed Algebras Induced by a Bounded Linear Functional". Khayyam Journal of Mathematics, 1, 1, 2015, 107-114. doi: 10.22034/kjm.2015.12290
HARVARD
Khoddami, A. R. (2015). 'Strongly Zeo-product Preserving Maps on Normed Algebras Induced by a Bounded Linear Functional', Khayyam Journal of Mathematics, 1(1), pp. 107-114. doi: 10.22034/kjm.2015.12290
VANCOUVER
Khoddami, A. R. Strongly Zeo-product Preserving Maps on Normed Algebras Induced by a Bounded Linear Functional. Khayyam Journal of Mathematics, 2015; 1(1): 107-114. doi: 10.22034/kjm.2015.12290