id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-1598	Dutta, Tapan K.; Shum, Kar Ping; Mandal, Shobhan	Singular Ideals of Ternary Semirings	2012	13	.pdf	application/pdf	6952	581	88	M is called a right ternary semimodule over a ternary semiring S or simply a right ternary S-semimodule if there exists a mapping M × S × S −→ M (images to be denoted by ms1s2 for all m ∈ M and s1, s2 ∈ S) satisfying the following conditions: (i) (m1 +m2)s1s2 = m1s1s2 +m2s1s2 (ii) m1s1(s2 + s3) = m1s1s2 +m1s1s3 (iii) m1(s1 + s2)s3 = m1s1s3 +m1s2s3 (iv) (m1s1s2)s3s4 = m1(s1s2s3)s4 = m1s1(s2s3s4) (v) 0M s1s2 = 0M = m1s10S = m10Ss2 for all m1, m2 ∈ M and for all s1, s2, s3, s4 ∈ S. A left ternary S-semimodule can be similarly defined. As h1s3s ∈ H for all s ∈ S, h1s3s ∈ rS(s1as2) ∩ H for all s ∈ S. Now, h1s3s = 0 for all s ∈ S ⇒ s2h1s3st = 0 for all s, t ∈ S ⇒ S ⊆ rS(s2h1s3) i.e. rS(s2h1s3) = S. Now by condition (α) this implies that s2h1s3 = 0, a contradiction.	cache/ejpam-1598.pdf	txt/ejpam-1598.txt