arXiv:2003.05317v2 [math.RA] 12 Mar 2020 NONSURJECTIVE ZERO PRODUCT PRESERVERS BETWEEN MATRICES OVER AN ARBITRARY FIELD CHI-KWONG LI, MING-CHENG TSAI, YA-SHU WANG AND NGAI-CHING WONG Abstract. In this paper, we give concrete descriptions of additive or linear disjointness pre- servers between matrix algebras over an arbitrary field F of different sizes. In particular, we show that a linear map Φ : Mn(F) → Mr (F) preserving zero products carries the form Φ(A)= S  R ⊗ A 0 0 Φ0(A)  S −1 , for some invertible matrices R in M k (F), S in Mr (F) and a zero product preserving linear map Φ0 : Mn(F) → M r−nk (F) with range consisting of nilpotent matrices. Here, either R or Φ0 can be vacuous. The structure of Φ0 could be quite arbitrary. We classify Φ0 with some additional assumption. When Φ(In) has a zero nilpotent part, especially when Φ(In) is diagonalizable, we have Φ0(X)Φ0(Y ) = 0 for all X, Y in Mn(F), and we give more information about Φ0 in this case. Similar results for double zero product preservers and orthogonality preservers are obtained. 1. Introduction There are considerable interests in studying preserver problems for matrices or operators; see, for example, [6, 13, 15–18, 21, 23, 24], and the references therein. Many preserver problems are connected to the study in those maps Φ of matrices or operators preserving zero products, i.e., Φ(A)Φ(B)=0 whenever AB =0. See, for example, [1, 3–5, 7, 12, 14]. It is usually expected that Φ gives rise to an algebra or a Jordan homomorphism. Most studies focus on surjective linear maps because general maps may not have nice structure. Even for (necessarily nonsurjective) linear preservers between two matrix algebras of different sizes, the results can be very complicated and intractable. Denote by M n = M n (F) the algebra of n × n matrices over a field F. The classical results of Jacobson, Rickart, Kaplansky, Herstien, etc. (see, e.g., [9, 10]), together with the Skolem-Noether theorem, ensure that every surjective zero product preserving linear map Φ : M n → M r is a scalar multiple of an inner algebra isomorphism, A → αS −1 AS , for a nonzero scalar α and an invertible S in M n (and thus n = r). See, e.g., [6, Theorems 2.6 and 3.1]. Date : July 24, 2020. 2000 Mathematics Subject Classification. 08A35, 15A86, 46L10, 46L40, 47B48. Key words and phrases. ring, Jordan, and algebra homomorphisms; zero product, orthogonality and idempo- tent preservers; matrix algebras. 1