The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 7(1) (2021), Pages 129-135 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: July 25, 2021 Reviewed: August 26, 2021 Accepted: October 11, 2021 DOI: https://doi.org/10.18860/ca.v7i1.13001 The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings Ahmad Faisol1, Fitriani2 1,2Department of Mathematics, Faculty of Mathematics and Natural Sciences Universitas Lampung, Bandar Lampung Email: ahmadfaisol@fmipa.unila.ac.id, fitriani.1984@fmipa.unila.ac.id ABSTRACT Let 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) be a matrix rings over skew generalized power series rings, where 𝑅1, 𝑅2 are commutative rings with an identity element, (𝑆1, ≀1), (𝑆2, ≀2) are strictly ordered monoids, πœ”1: 𝑆1 β†’ 𝐸𝑛𝑑(𝑅1), πœ”2: 𝑆2 β†’ 𝐸𝑛𝑑(𝑅2) are monoid homomorphisms. In this research, we define a mapping 𝜏 from 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) to 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) by using a strictly ordered monoid homomorphism 𝛿: (𝑆1, ≀1) β†’ (𝑆2, ≀2), and ring homomorphisms πœ‡: 𝑅1 β†’ 𝑅2 and 𝜎: 𝑅1[[𝑆1, ≀1, πœ”1]] β†’ 𝑅2[[𝑆2, ≀2, πœ”2]]. Furthermore, we prove that 𝜏 is a ring homomorphism, and also we give the sufficient conditions for 𝜏 to be a monomorphism, epimorphism, and isomorphism. Keywords: matrix rings; homomorphisms; skew generalized power series rings. INTRODUCTION In [1], it has been explained that a matrix is an arrangement of mathematical objects in rectangular rows and columns enclosed by square brackets or regular brackets. These mathematical objects are commonly called entries. If the matrix entries are members of a ring, the matrix is called the matrix over the ring [2]. A ring is a non- empty set with two binary operations and satisfies several axioms [3]. The skew generalized power series rings (SGPSR) 𝑅[[𝑆, ≀, πœ”]] is one example of a ring [4]. This ring is defined as the set of all functions 𝑓 from a strictly ordered monoid (𝑆, ≀) to a ring 𝑅 with an identity element, that supp(𝑓) is Artinian and narrow, with pointwise addition operation and convolution multiplication operation using a monoid homomorphism πœ”: 𝑆 β†’ End(𝑅). Some research related to the properties of SGPSR 𝑅[[𝑆, ≀, πœ”]], can be seen in Mazurek et al. [5]-[10] and Faisol et al. [11]-[16]. A set of matrices over a ring that forms a ring under matrix addition and matrix multiplication is called a matrix ring [17]. Furthermore, the set of all 𝑛 Γ— 𝑛 matrices with entries in ring 𝑅 is a matrix ring denoted by 𝑀𝑛 (𝑅). In 2021, Rugayah et al. [18] have constructed the set of all matrices over SGPSR 𝑅[[𝑆, ≀, πœ”]], denoted by 𝑀𝑛 (𝑅[[𝑆, ≀, πœ”]]). Moreover, they have defined the ideal of matrix ring over SGPSR 𝑅[[𝑆, ≀, πœ”]] and studied its ideal properties. One of the essential concepts in the ring structure is a ring homomorphism, a mapping from ring to ring that preserves binary operations on these rings. In [19], the matrix ring homomorphism from 𝑀𝑛(𝑅1) to 𝑀𝑛 (𝑅2) defined by 𝜎([π‘Žπ‘–π‘— ]) = [πœ‡(π‘Žπ‘–π‘— )] for https://doi.org/10.18860/ca.v7i1.13001 mailto:ahmadfaisol@fmipa.unila.ac.id mailto:fitriani.1984@fmipa.unila.ac.id The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings Ahmad Faisol 130 every π‘Žπ‘–π‘— ∈ 𝑅1 where πœ‡: 𝑅1 β†’ 𝑅2 is a ring homomorphism has constructed. Several studies related to matrix ring homomorphism can be seen in [20],[21]. This construction motivates us to study the ring homomorphism on the ring matrix over SGPSR 𝑅[[𝑆, ≀ , πœ”]]. Therefore, in this research, matrix rings over the SGPSR 𝑅[[𝑆, ≀, πœ”]] were constructed, i.e., 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) where 𝑅1, 𝑅2 are rings, (𝑆1, ≀1), (𝑆2, ≀2) are strictly ordered monoids, and πœ”1: 𝑆1 β†’ 𝐸𝑛𝑑(𝑅1), πœ”2: 𝑆2 β†’ 𝐸𝑛𝑑(𝑅2) are monoid homomorphisms. Next, the maping 𝜏 from 𝑀𝑛(𝑅1[[𝑆1, ≀1, πœ”1]]) to 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) is defined by using a strictly ordered monoid homomorphism 𝛿: (𝑆1, ≀1) β†’ (𝑆2, ≀2), and ring homomorphisms πœ‡: 𝑅1 β†’ 𝑅2 and 𝜎: 𝑅1[[𝑆1, ≀1, πœ”1]] β†’ 𝑅2[[𝑆2, ≀2, πœ”2]]. Furthermore, it is proved that 𝜏 is a matrix ring homomorphism, and the sufficient conditions for 𝜏 to be a monomorphism, epimorphism, and isomorphism are also given. METHODS The method used in this research is a literature study from books and scientific journals. The following steps can be obtained in the results. We construct the matrix rings over SGPSR 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]), where 𝑅1, 𝑅2 are given rings, strictly ordered monoid (𝑆1, ≀1), (𝑆2, ≀2), strictly ordered monoid homomorphism 𝛿: (𝑆1, ≀1) β†’ (𝑆2, ≀2), and monoid homomorphisms πœ”1: 𝑆1 β†’ End(𝑅1), πœ”2: 𝑆2 β†’ End(𝑅2). Next, we define a mapping Ο„ from Mn(R1[[S1, ≀1, Ο‰1]]) to Mn(R2[[S2, ≀2, Ο‰2]]), by using a strictly ordered monoid homomorphism 𝛿: (𝑆1, ≀1) β†’ (𝑆2, ≀2), ring homomorphisms πœ‡: 𝑅1 β†’ 𝑅2 and 𝜎: 𝑅1[[𝑆1, ≀1, πœ”1]] β†’ 𝑅2[[𝑆2, ≀2, πœ”2]]. Furthermore, we prove that Ο„ is a ring homomorphism. Finally, we give sufficient conditions for Ο„ to be a monomorphism, epimorphism, and isomorphism. RESULTS AND DISCUSSION Mazurek and Ziembowski [4] give the structure of skew generalized power series rings (SGPSR) as follows. Let 𝑅1, 𝑅2 are rings, (𝑆1, ≀1), (𝑆2, ≀2) are strictly ordered monoids, and πœ”1: 𝑆1 β†’ 𝐸𝑛𝑑(𝑅1), πœ”2: 𝑆2 β†’ 𝐸𝑛𝑑(𝑅2) are monoid homomorphisms. Homomorphic image of πœ”1 and πœ”2 are denoted by πœ”1 𝑠 and πœ”2 𝑒 for all 𝑠 ∈ 𝑆1 and ∈ 𝑆2 . Therefore, πœ”1 𝑠+𝑑 = πœ”1(𝑠 + 𝑑) = πœ”1(𝑠) + πœ”1(𝑑) = πœ”1 𝑠 + πœ”1 𝑑 , (1) and πœ”2 𝑒+𝑣 = πœ”2(𝑒 + 𝑣) = πœ”2(𝑒) + πœ”2(𝑣) = πœ”2 𝑒 + πœ”2 𝑣 , (2) for every all 𝑠, 𝑑 ∈ 𝑆1 and 𝑒, 𝑣 ∈ 𝑆2. Next, let 𝑅1[[𝑆1, ≀1, πœ”1]] = {𝑓: 𝑆1 β†’ 𝑅1|supp(𝑓) Artinian and narrow} and 𝑅2[[𝑆2, ≀2, πœ”2]] = {𝛼: 𝑆2 β†’ 𝑅2|supp(𝛼) Artinian and narrow}, where supp(𝑓) = {𝑠 ∈ 𝑆1|𝑓(𝑠) β‰  0} and supp(𝛼) = {𝑒 ∈ 𝑆2|𝛼(𝑒) β‰  0}. Under pointwise addition and convolution multiplication defined by (𝑓 + 𝑔)(𝑠) = 𝑓(𝑠) + 𝑔(𝑠), (3) (𝛼 + 𝛽)(𝑒) = 𝛼(𝑒) + 𝛽(𝑒), (4) and (𝑓𝑔)(𝑠) = βˆ‘ 𝑓(π‘₯)πœ”1 π‘₯ (𝑔(𝑦))π‘₯+𝑦=𝑠 , (5) (𝛼𝛽)(𝑒) = βˆ‘ 𝛼(𝑝)πœ”2 𝑝 (𝛽(π‘ž))𝑝+π‘ž=𝑒 , (6) 𝑅1[[𝑆1, ≀1, πœ”1]] and 𝑅2[[𝑆2, ≀2, πœ”2]] be a skew generalized power series rings, for every 𝑠 ∈ 𝑆1, 𝑓, 𝑔 ∈ 𝑅1[[𝑆1, ≀1, πœ”1]], and 𝑒 ∈ 𝑆2, 𝛼, 𝛽 ∈ 𝑅2[[𝑆2, ≀2, πœ”2]]. The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings Ahmad Faisol 131 According to [22], a strictly ordered monoid homomorphism Ξ΄: (S1, ≀1) β†’ (S2, ≀2) is a monoid homomorphism such that if s <1 t, then Ξ΄(s) <2 Ξ΄(t) for every s, t ∈ S1. Now, let Ξ΄ be a monomorphism such that for any Artinian and narrow subset 𝑇 of S1, Ξ΄(𝑇) is an Artinian and narrow subset of S2, and ΞΌ: R1 β†’ R2 is a ring homomorphism such that for every s ∈ S1 the following diagram is commutative: 𝑆1 𝑓 ↓ 𝑅1 𝛿 β†’ πœ‡ β†’ 𝑆2 ↓ 𝛼 𝑅2 πœ”1 𝑠 ↓ ↻ ↓ πœ”2 𝛿(𝑠) 𝑅1 πœ‡ β†’ 𝑅2 Figure 1. Commutative diagram πœ”2 𝛿(𝑠) ∘ πœ‡ = πœ‡ ∘ πœ”1 𝑠 For 𝑓 ∈ R1[[S1, ≀1, Ο‰1]], let 𝛼: 𝑆2 β†’ 𝑅2 be the map defined as follows: 𝛼(𝑑) = { πœ‡ ∘ 𝑓 ∘ 𝛿 βˆ’1(𝑑) if 𝑑 ∈ 𝛿(𝑆1) 0 otherwise. (7) Since supp(𝛼) βŠ† 𝛿(supp(𝑓)), based on [23](1.(a)), 𝛼 ∈ 𝑅2[[𝑆2, ≀2, πœ”2]]. Therefore, we can define a map 𝜎: 𝑅1[[𝑆1, ≀1, πœ”1]] β†’ 𝑅2[[𝑆2, ≀2, πœ”2]] by putting 𝜎(𝑓) = 𝛼 in (7). According to [24](Lemma 8.1.6), the map Οƒ: R1[[S1, ≀1, Ο‰1]] β†’ R2[[S2, ≀2, Ο‰2]] is a ring homomorphism, and Ker(Οƒ) = (Ker(ΞΌ))[[S1, ≀1, Ο‰1]]. Now, we construct the matrix rings 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]), that are the sets of all matrices over SGPSR 𝑅1[[𝑆1, ≀1, πœ”1]] and 𝑅2[[𝑆2, ≀2, πœ”2]] defined by 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) = {[𝑓𝑖𝑗 ]|𝑓𝑖𝑗 ∈ 𝑅1[[𝑆1, ≀1, πœ”1]]; 𝑖, 𝑗 = 1, 2, β‹― , 𝑛}, (8) and 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) = {[𝛼𝑖𝑗 ]|𝛼𝑖𝑗 ∈ 𝑅2[[𝑆2, ≀2, πœ”2]]; 𝑖, 𝑗 = 1, 2, β‹― , 𝑛}, (9) with addition matrix operation [𝑓𝑖𝑗 ] + [𝑔𝑖𝑗 ] = [𝑓𝑖𝑗 + 𝑔𝑖𝑗 ] , (10) [𝛼𝑖𝑗 ] + [𝛽𝑖𝑗 ] = [𝛼𝑖𝑗 + 𝛽𝑖𝑗 ] , (11) and multiplication matrix operation [𝑓𝑖𝑗 ][𝑔𝑖𝑗 ] = [β„Žπ‘–π‘— ] , (12) [𝛼𝑖𝑗 ][𝛽𝑖𝑗 ] = [𝛾𝑖𝑗 ] , (13) where β„Žπ‘–π‘— = βˆ‘ π‘“π‘–π‘˜ π‘”π‘˜π‘— 𝑛 π‘˜=1 dan 𝛾𝑖𝑗 = βˆ‘ π›Όπ‘–π‘˜ π›½π‘˜π‘— 𝑛 π‘˜=1 , for every [𝑓𝑖𝑗 ], [𝑔𝑖𝑗 ] ∈ 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) and [𝛼𝑖𝑗 ], [𝛽𝑖𝑗 ] ∈ 𝑀𝑛(𝑅2𝑅2[[𝑆2, ≀2, πœ”2]]). For every [𝑓𝑖𝑗 ] ∈ 𝑀𝑛(𝑅1[[𝑆1, ≀1, πœ”1]]), we define the map 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) β†’ 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) by 𝜏([𝑓𝑖𝑗 ]) = [𝜎(𝑓𝑖𝑗 )], (14) for every [𝑓𝑖𝑗 ] ∈ 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]). The following theorem shows that 𝜏 is a ring homomorphism. The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings Ahmad Faisol 132 Proposition 1 Let 𝑀𝑛(𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≀2, πœ”2]]) be matrix rings over SGPSR. The mapping 𝜏 that is defined in (14) is a ring homomorphism. Proof Based on [24](Lemma 8.1.6), for i, j = 1, 2, β‹― , n, there is 𝛼𝑖𝑗 = 𝜎(𝑓𝑖𝑗 ) ∈ R2[[𝑆2, ≀2, πœ”2]] for every 𝑓𝑖𝑗 ∈ 𝑅1[[𝑆1, ≀1, πœ”1]]. Therefore, 𝜏([𝑓𝑖𝑗 ]) = [𝜎(𝑓𝑖𝑗 )] = [𝛼𝑖𝑗 ] is well-defined. For any 𝑑 ∈ 𝑆2, 𝑓𝑖𝑗 , 𝑔𝑖𝑗 ∈ 𝑅1[[𝑆1, ≀1, πœ”1]], we have πœ‡ ∘ (𝑓𝑖𝑗 + 𝑔𝑖𝑗 ) ∘ 𝛿 βˆ’1(𝑑) = πœ‡ ((𝑓𝑖𝑗 + 𝑔𝑖𝑗 )(𝛿 βˆ’1(𝑑))) = πœ‡ (𝑓𝑖𝑗 (𝛿 βˆ’1(𝑑)) + 𝑔𝑖𝑗 (𝛿 βˆ’1(𝑑))) = πœ‡ (𝑓𝑖𝑗 (𝛿 βˆ’1(𝑑))) + πœ‡ (𝑔𝑖𝑗 (𝛿 βˆ’1(𝑑))) = (πœ‡ ∘ 𝑓𝑖𝑗 ∘ 𝛿 βˆ’1)(𝑑) + (πœ‡ ∘ 𝑔𝑖𝑗 ∘ 𝛿 βˆ’1)(𝑑), and πœ‡ ∘ (𝑓𝑖𝑗 𝑔𝑖𝑗 ) ∘ 𝛿 βˆ’1(𝑑) = πœ‡ ((𝑓𝑖𝑗 𝑔𝑖𝑗 )(𝛿 βˆ’1(𝑑))) = πœ‡ ( βˆ‘ 𝑓𝑖𝑗 (π‘₯)πœ”1 π‘₯ (𝑔𝑖𝑗 (𝑦)) π‘₯+𝑦=π›Ώβˆ’1(𝑑) ) = βˆ‘ πœ‡ (𝑓𝑖𝑗 (π‘₯)πœ”1 π‘₯ (𝑔𝑖𝑗 (𝑦))) π‘₯+𝑦=π›Ώβˆ’1(𝑑) = βˆ‘ πœ‡ (𝑓𝑖𝑗 (π‘₯)) πœ‡ (πœ”1 π‘₯ (𝑔𝑖𝑗 (𝑦))) π‘₯+𝑦=π›Ώβˆ’1(𝑑) = βˆ‘ πœ‡ (𝑓𝑖𝑗 (π‘₯)) πœ”2 𝛿(π‘₯) (πœ‡ (𝑔𝑖𝑗 (𝑦))) π‘₯+𝑦=π›Ώβˆ’1(𝑑) 𝛿(π‘₯)+𝛿(𝑦)=𝑑 = βˆ‘ πœ‡ (𝑓𝑖𝑗 (𝛿 βˆ’1(𝑒))) πœ”2 𝑒 (πœ‡ (𝑔𝑖𝑗 (𝛿 βˆ’1(𝑣)))) 𝑒+𝑣=𝑑 = βˆ‘ (πœ‡ ∘ 𝑓𝑖𝑗 ∘ 𝛿 βˆ’1)(𝑒)πœ”2 𝑒 ((πœ‡ ∘ 𝑔𝑖𝑗 ∘ 𝛿 βˆ’1)(𝑣)) 𝑒+𝑣=𝑑 = (πœ‡ ∘ 𝑓𝑖𝑗 ∘ 𝛿 βˆ’1)(πœ‡ ∘ 𝑔𝑖𝑗 ∘ 𝛿 βˆ’1)(𝑑). In other words, πœ‡ ∘ (𝑓𝑖𝑗 + 𝑔𝑖𝑗 ) ∘ 𝛿 βˆ’1 = (πœ‡ ∘ 𝑓𝑖𝑗 ∘ 𝛿 βˆ’1) + (πœ‡ ∘ 𝑔𝑖𝑗 ∘ 𝛿 βˆ’1) and πœ‡ ∘ (𝑓𝑖𝑗 𝑔𝑖𝑗 ) ∘ 𝛿 βˆ’1 = (πœ‡ ∘ 𝑓𝑖𝑗 ∘ 𝛿 βˆ’1)(πœ‡ ∘ 𝑔𝑖𝑗 ∘ 𝛿 βˆ’1) for every 𝑓ij, 𝑔ij ∈ 𝑅1[[𝑆1, ≀1, πœ”1]]. Now, we prove that Ο„ is a ring homomorphism. For any [𝑓𝑖𝑗 ], [𝑔𝑖𝑗 ] ∈ Mn(𝑅1[[𝑆1, ≀1, πœ”1]]), we obtain: (i) Ο„([𝑓𝑖𝑗 ] + [𝑔𝑖𝑗 ]) = Ο„([𝑓𝑖𝑗 + 𝑔𝑖𝑗 ]) = [𝜎(𝑓𝑖𝑗 + 𝑔𝑖𝑗 )] = [πœ‡ ∘ (𝑓𝑖𝑗 + 𝑔𝑖𝑗 ) ∘ 𝛿 βˆ’1] = [(ΞΌ ∘ 𝑓𝑖𝑗 ∘ Ξ΄ βˆ’1) + (ΞΌ ∘ 𝑔𝑖𝑗 ∘ Ξ΄ βˆ’1)] = [ΞΌ ∘ 𝑓𝑖𝑗 ∘ Ξ΄ βˆ’1] + [ΞΌ ∘ 𝑔𝑖𝑗 ∘ Ξ΄ βˆ’1] = [𝜎(𝑓𝑖𝑗 )] + [𝜎(𝑔𝑖𝑗 )] = 𝜏([𝑓𝑖𝑗 ]) + 𝜏([𝑔𝑖𝑗 ]). The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings Ahmad Faisol 133 (ii) 𝜏([𝑓𝑖𝑗 ][𝑔𝑖𝑗 ]) = 𝜏([βˆ‘ π‘“π‘–π‘˜ π‘”π‘˜π‘— 𝑛 π‘˜=1 ]) = [𝜎 (βˆ‘ π‘“π‘–π‘˜ π‘”π‘˜π‘— 𝑛 π‘˜=1 )] = [πœ‡ ∘ (βˆ‘ π‘“π‘–π‘˜ π‘”π‘˜π‘— 𝑛 π‘˜=1 ) ∘ 𝛿 βˆ’1] = [βˆ‘ πœ‡ ∘ (π‘“π‘–π‘˜ π‘”π‘˜π‘— ) ∘ 𝛿 βˆ’1 𝑛 π‘˜=1 ] = [βˆ‘(πœ‡ ∘ π‘“π‘–π‘˜ ∘ 𝛿 βˆ’1)(πœ‡ ∘ π‘”π‘˜π‘— ∘ 𝛿 βˆ’1) 𝑛 π‘˜=1 ] = [πœ‡ ∘ 𝑓𝑖𝑗 ∘ 𝛿 βˆ’1][πœ‡ ∘ 𝑔𝑖𝑗 ∘ 𝛿 βˆ’1] = [𝜎(𝑓𝑖𝑗 )][𝜎(𝑔𝑖𝑗 )] =𝜏([𝑓𝑖𝑗 ])𝜏([𝑔𝑖𝑗 ]) According to (i) and (ii), it is proved that Ο„ is a ring homomorphism. ∎ The following proposition shows that Ker(𝜏) = 𝑀𝑛 ((Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]). Proposition 2 Let 𝑀𝑛(𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≀2, πœ”2]]) be matrix rings over SGPSR. Let 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) β†’ 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) is the map that is defined in (14). Then, Ker(𝜏) = 𝑀𝑛 ((Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]). Proof For any [𝑓𝑖𝑗 ] ∈ Ker(𝜏), we have 𝜏([𝑓𝑖𝑗 ]) = [𝜎(𝑓𝑖𝑗 )] = [0]. Therefore, for i, j = 1, 2, β‹― , n , 𝜎(𝑓𝑖𝑗 ) = 0. So, 𝑓𝑖𝑗 ∈ Ker(Οƒ). Based on [24](Lemma 8.1.6), Ker(𝜎) = (Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]. Therefore, 𝑓𝑖𝑗 ∈ (Ker(πœ‡))[[𝑆1, ≀1, πœ”1]] for all i, j = 1, 2, β‹― , n. So, [𝑓𝑖𝑗 ] ∈ 𝑀𝑛 ((Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]). Then, we get Ker(𝜏) βŠ‚ 𝑀𝑛 ((Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]). On the other side, for any [𝑓𝑖𝑗 ] ∈ 𝑀𝑛 ((Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]), we have 𝑓𝑖𝑗 ∈ (Ker(πœ‡))[[𝑆1, ≀1, πœ”1]] for all i, j = 1, 2, β‹― , n. According to [24](Lemma 8.1.6), Ker(𝜎) = (Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]. Therefore, 𝑓𝑖𝑗 ∈ Ker(𝜎). Then, 𝜎(𝑓𝑖𝑗 ) = 0 for all i, j = 1, 2, β‹― , n. So, we get [𝜎(𝑓𝑖𝑗 )] = [0] = 𝜏([𝑓𝑖𝑗 ]). In other words, [𝑓𝑖𝑗 ] ∈ Ker(𝜏). Hence, 𝑀𝑛 ((Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]) βŠ‚ Ker(𝜏). So, it is proved that Ker(𝜏) = 𝑀𝑛 ((Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]) ∎ Next, we give sufficient conditions for 𝜏 to be a monomorphism, epimorphism, and isomorphism. Proposition 3 Let 𝑀𝑛(𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≀2, πœ”2]]) be matrix rings over SGPSR. Let 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) β†’ 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) is the map that is defined in (14). If 𝛿 is an isomorphism and πœ‡ is a monomorphism, then 𝜏 is a monomorphism. Proof Based on Proposition 1, it is clear that 𝜏 is a ring homomorphism. So, we only have to show that 𝜏 is injective. If Ο„([𝑓𝑖𝑗 ]) = Ο„([𝑔𝑖𝑗 ]), then [𝜎(𝑓𝑖𝑗 )] = [𝜎(𝑔𝑖𝑗 )]. Hence, [ΞΌ ∘ 𝑓𝑖𝑗 ∘ Ξ΄ βˆ’1] = [ΞΌ ∘ 𝑔𝑖𝑗 ∘ Ξ΄ βˆ’1]. Therefore, we get ΞΌ ∘ 𝑓𝑖𝑗 ∘ Ξ΄ βˆ’1 = ΞΌ ∘ 𝑔𝑖𝑗 ∘ Ξ΄ βˆ’1 for all i, j = The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings Ahmad Faisol 134 1, 2, β‹― , n. In other words, for any t ∈ S2, we have ΞΌ (𝑓𝑖𝑗 (Ξ΄ βˆ’1(t))) = ΞΌ (𝑔𝑖𝑗 (Ξ΄ βˆ’1(t))). Since ΞΌ is a monomorphism, 𝑓𝑖𝑗 (Ξ΄ βˆ’1(t)) = 𝑔𝑖𝑗 (Ξ΄ βˆ’1(t)). Since Ξ΄ is an isomorphism, 𝑓𝑖𝑗 (s) = 𝑔𝑖𝑗 (s) for every s ∈ S1. So, 𝑓𝑖𝑗 = 𝑔𝑖𝑗 for all 𝑖, 𝑗 = 1, 2, β‹― , n. Therefore [𝑓𝑖𝑗 ] = [𝑔𝑖𝑗 ]. So, it is proved that if Ο„([𝑓𝑖𝑗 ]) = Ο„([𝑔𝑖𝑗 ]), then [𝑓𝑖𝑗 ] = [𝑔𝑖𝑗 ]. Hence, Ο„ is injective. ∎ Proposition 4 Let 𝑀𝑛(𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≀2, πœ”2]]) be matrix rings over SGPSR. Let 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) β†’ 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) is the map that is defined in (14). If 𝜎 is an epimorphism, then 𝜏 is an epimorphism. Proof Based on Proposition 1, 𝜏 is a ring homomorphism. So, we only have to show that 𝜏 is surjective. In other words, we have to prove that Im(𝜏) = 𝑀𝑛(𝑅2[[𝑆2, ≀2, πœ”2]]). It is clear that Im(𝜏) βŠ‚ 𝑀𝑛(𝑅2[[𝑆2, ≀2, πœ”2]]), so it suffices to show that 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) βŠ‚ Im(𝜏). For any [Ξ±ij] ∈ Mn(𝑅2[[𝑆2, ≀2, πœ”2]]), then α𝑖𝑗 ∈ 𝑅2[[𝑆2, ≀2, πœ”2]] for all 𝑖, 𝑗 = 1, 2, β‹― , 𝑛. Since 𝜎 is an epimorphism, there is 𝑓𝑖𝑗 ∈ 𝑅1[[𝑆1, ≀1, πœ”1]] such that 𝜎(𝑓𝑖𝑗 ) = α𝑖𝑗 . Therefore, there is [𝑓𝑖𝑗 ] ∈ 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) such that [α𝑖𝑗 ] = [𝜎(𝑓𝑖𝑗 )] = Ο„([𝑓𝑖𝑗 ]) for every [α𝑖𝑗 ] ∈ 𝑀𝑛(𝑅2[[𝑆2, ≀2, πœ”2]]). So, [α𝑖𝑗 ] ∈ Im(𝜏). In other words, 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) βŠ‚ Im(𝜏). Hence, that 𝜏 is surjective. ∎ Corollary 5 Let 𝑀𝑛(𝑅1[[𝑆1, ≀1, πœ”1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≀2, πœ”2]]) be matrix rings over SGPSR. Let 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) β†’ 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) is the map that is defined in (14). If 𝛿 is an isomorphism, πœ‡ is a monomorphism, and 𝜎 is an epimorphism, then 𝜏 is an isomorphism. CONCLUSIONS A ring homomorphism 𝜏 from the matrix ring 𝑀𝑛 (𝑅1[[𝑆1, ≀1, πœ”1]]) to the matrix ring 𝑀𝑛 (𝑅2[[𝑆2, ≀2, πœ”2]]) can be constructed by using a strictly ordered monoid homomorphism 𝛿: (𝑆1, ≀1) β†’ (𝑆2, ≀2), and ring homomorphisms πœ‡: 𝑅1 β†’ 𝑅2 and 𝜎: 𝑅1[[𝑆1, ≀1, πœ”1]] β†’ 𝑅2[[𝑆2, ≀2, πœ”2]]. Furthermore, it also proves that Ker(𝜏) is equal to the matrix ring over SGPSR (Ker(πœ‡))[[𝑆1, ≀1, πœ”1]]. Moreover, if 𝛿 is an isomorphism and πœ‡ is a monomorphism, then 𝜏 is a monomorphism. While, if 𝜎 is an epimorphism, then 𝜏 is an epimorphism. Consequently, 𝜏 is an isomorphism if 𝛿 is an isomorphism, πœ‡ is a monomorphism, and 𝜏 is an epimorphism. 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