id sid eid entity type cana-1295 1 1 1074 CARDINAL cana-1295 2 1 7s CARDINAL cana-1295 2 2 2024 CARDINAL cana-1295 2 3 203 CARDINAL cana-1295 2 4 1n CARDINAL cana-1295 3 1 sandhya rani PERSON cana-1295 3 2 2 CARDINAL cana-1295 3 3 g. srinivasa rao PERSON cana-1295 3 4 3ch CARDINAL cana-1295 4 1 1research CARDINAL cana-1295 4 2 mathematics & statistics ORG cana-1295 4 3 guntur GPE cana-1295 4 4 a.p. GPE cana-1295 4 5 india GPE cana-1295 4 6 n.sandhayarani@rguktrkv.ac.in 1mathematics DATE cana-1295 4 7 idupulapaya GPE cana-1295 4 8 a.p. GPE cana-1295 4 9 india GPE cana-1295 5 1 2associate CARDINAL cana-1295 5 2 department of mathematics & statistics ORG cana-1295 5 3 guntur GPE cana-1295 5 4 a.p. GPE cana-1295 6 1 3associate CARDINAL cana-1295 6 2 guntur GPE cana-1295 6 3 a.p. GPE cana-1295 6 4 01-06-2024 CARDINAL cana-1295 6 5 03-07-2024 CARDINAL cana-1295 6 6 29-07-2024 CARDINAL cana-1295 8 1 two CARDINAL cana-1295 9 1 1994 to 1996 DATE cana-1295 11 1 vasantha kandaswamy PERSON cana-1295 11 2 1997 DATE cana-1295 14 1 vasantha kandaswamy PERSON cana-1295 14 2 bi-algebraic framework.agboola PERSON cana-1295 14 3 akinola ORG cana-1295 15 1 algebraic frame PERSON cana-1295 16 1 g. srinivasa rao et.al[1-2 PERSON cana-1295 16 2 13 CARDINAL cana-1295 17 1 1 CARDINAL cana-1295 17 2 1.1 CARDINAL cana-1295 17 3 βˆ— CARDINAL cana-1295 17 4 algebraic structure(𝐺,βˆ— PERSON cana-1295 18 1 βˆ— CARDINAL cana-1295 18 2 1074 CARDINAL cana-1295 19 1 7s CARDINAL cana-1295 19 2 2024 CARDINAL cana-1295 19 3 204 CARDINAL cana-1295 19 4 1.2 CARDINAL cana-1295 19 5 𝐺and β€² βˆ— β€² ORG cana-1295 19 6 algebraic structure(𝐺,βˆ— PERSON cana-1295 20 1 𝑖)π‘Ž ORG cana-1295 20 2 𝑖𝑖)π‘Ž ORG cana-1295 20 3 1.3 CARDINAL cana-1295 22 1 π‘₯1π‘₯2(π‘₯3π‘₯4π‘₯5 ORDINAL cana-1295 22 2 ∈ ORG cana-1295 22 3 1.4 CARDINAL cana-1295 22 4 two CARDINAL cana-1295 24 1 two CARDINAL cana-1295 24 2 𝐺1 CARDINAL cana-1295 25 1 𝑖𝑖𝑖)(𝐺1,β‹… ORG cana-1295 25 2 1.5 CARDINAL cana-1295 26 1 1.6 CARDINAL cana-1295 26 2 two CARDINAL cana-1295 28 1 two CARDINAL cana-1295 28 2 𝐺1 CARDINAL cana-1295 29 1 𝐺1 CARDINAL cana-1295 29 2 𝐺2 𝑖𝑖)(𝐺1 ORG cana-1295 30 1 𝑖𝑖𝑖)(𝐺1,β‹… ORG cana-1295 31 1 1.7 CARDINAL cana-1295 31 2 1 CARDINAL cana-1295 31 3 βˆ’1 DATE cana-1295 31 4 βˆ’π‘–},βˆ™ ORG cana-1295 32 1 2 CARDINAL cana-1295 32 2 2.1 CARDINAL cana-1295 32 3 𝑇1 βˆͺ MONEY cana-1295 33 1 𝑇1 CARDINAL cana-1295 36 1 π‘₯1π‘₯2(π‘₯3π‘₯4π‘₯5 ORDINAL cana-1295 37 1 ∈ ORG cana-1295 37 2 𝑇1 PRODUCT cana-1295 39 1 π‘Žπ‘(𝑐𝑑𝑒)]βˆ€π‘Ž NORP cana-1295 39 2 𝑏 ORG cana-1295 39 3 𝑇1 βˆͺ MONEY cana-1295 39 4 𝑇1 & 𝑇2 ORG cana-1295 40 1 2.2 CARDINAL cana-1295 40 2 0 CARDINAL cana-1295 40 3 1 CARDINAL cana-1295 40 4 1 CARDINAL cana-1295 40 5 0 CARDINAL cana-1295 40 6 0 CARDINAL cana-1295 40 7 2.3 CARDINAL cana-1295 41 1 2.4 CARDINAL cana-1295 42 1 1074 CARDINAL cana-1295 43 1 7s CARDINAL cana-1295 43 2 2024 CARDINAL cana-1295 43 3 205 CARDINAL cana-1295 43 4 2.5 CARDINAL cana-1295 43 5 𝑇1 βˆͺ MONEY cana-1295 43 6 𝑇1 CARDINAL cana-1295 43 7 𝐢,βˆ™ GPE cana-1295 43 8 3𝒁 CARDINAL cana-1295 44 1 2.6 CARDINAL cana-1295 44 2 𝑆2 DATE cana-1295 44 3 π‘Ž1𝑏1𝑐1βˆ€π‘Ž1 PERSON cana-1295 44 4 ∈ PRODUCT cana-1295 44 5 π‘Ž2𝑏2𝑐2βˆ€π‘Ž2 GPE cana-1295 44 6 𝑏2 GPE cana-1295 44 7 2.7 CARDINAL cana-1295 44 8 𝑆2 DATE cana-1295 45 1 βŠ† 𝑆1 𝑆2𝑆2𝑆2 βŠ† QUANTITY cana-1295 45 2 2.8 CARDINAL cana-1295 45 3 𝑇1 βˆͺ MONEY cana-1295 45 4 𝑇1 CARDINAL cana-1295 45 5 𝐢,βˆ™ GPE cana-1295 45 6 two CARDINAL cana-1295 46 1 𝑆2 DATE cana-1295 46 2 𝑆1 CARDINAL cana-1295 46 3 𝑆2 CARDINAL cana-1295 46 4 3𝒁 CARDINAL cana-1295 46 5 two CARDINAL cana-1295 46 6 𝑇.so,𝑆 βŠ‚ 𝑇 ORG cana-1295 46 7 𝑇. NORP cana-1295 46 8 2.9 CARDINAL cana-1295 46 9 two CARDINAL cana-1295 46 10 𝑇. NORP cana-1295 47 1 𝑆1 CARDINAL cana-1295 47 2 𝑆2 CARDINAL cana-1295 47 3 two CARDINAL cana-1295 47 4 𝑇. NORP cana-1295 47 5 ∈ 𝑆1 PRODUCT cana-1295 47 6 ∈ 𝑆1 PRODUCT cana-1295 47 7 ∈ 𝑆1 PRODUCT cana-1295 47 8 𝑆1 CARDINAL cana-1295 47 9 𝑆2 DATE cana-1295 47 10 thatπ‘Žπ‘π‘ PERSON cana-1295 47 11 ∈ ORG cana-1295 47 12 𝑆1 PRODUCT cana-1295 47 13 𝑆1 CARDINAL cana-1295 47 14 𝑇. NORP cana-1295 47 15 2.10 CARDINAL cana-1295 47 16 𝑇. NORP cana-1295 47 17 π‘†π›Όπ›Όβˆˆβˆ† DATE cana-1295 48 1 ∈ ORG cana-1295 48 2 π‘†π›Όπ›Όβˆˆβˆ† DATE cana-1295 48 3 ∈ 𝑆𝛼 ORG cana-1295 48 4 ∈ ORG cana-1295 48 5 ∈ 𝑆𝛼 ORG cana-1295 48 6 𝑆𝛼 PERSON cana-1295 48 7 ∈ 𝑆𝛼 ORG cana-1295 48 8 ∈ 𝑆𝛼 ORG cana-1295 48 9 ∈ ORG cana-1295 48 10 ∈ ORG cana-1295 48 11 π‘†π›Όπ›Όβˆˆβˆ† DATE cana-1295 48 12 𝑇. NORP cana-1295 48 13 2.11 CARDINAL cana-1295 48 14 & 𝐴 ORG cana-1295 48 15 𝑇. NORP cana-1295 48 16 t containing a. DATE cana-1295 48 17 βˆ†β‰  CARDINAL cana-1295 48 18 π‘†π‘ βˆˆβˆ† CARDINAL cana-1295 48 19 𝐴 βŠ† ORG cana-1295 48 20 1074 CARDINAL cana-1295 49 1 7s CARDINAL cana-1295 49 2 2024 CARDINAL cana-1295 49 3 206 CARDINAL cana-1295 49 4 𝑇. NORP cana-1295 49 5 π‘†βˆ— βŠ† MONEY cana-1295 50 1 2.12 CARDINAL cana-1295 50 2 3.1 CARDINAL cana-1295 50 3 𝐴1 PRODUCT cana-1295 50 4 𝑇1 CARDINAL cana-1295 50 5 𝑇1 PRODUCT cana-1295 50 6 𝐴1 PRODUCT cana-1295 50 7 ∈ ORG cana-1295 50 8 𝐴1 PRODUCT cana-1295 50 9 ∈ ORG cana-1295 51 1 3.2 CARDINAL cana-1295 51 2 𝑇𝑇𝐴 βŠ† 𝐴. ORG cana-1295 52 1 𝑇1 βˆͺ 𝑇2 MONEY cana-1295 52 2 𝑇1 CARDINAL cana-1295 52 3 ∈ ORG cana-1295 52 4 𝑇. 𝐼 ORG cana-1295 52 5 𝑇1 ∩ 𝑇2respectively ORG cana-1295 53 1 𝐼 ORG cana-1295 53 2 𝑇. NORP cana-1295 53 3 3.4 CARDINAL cana-1295 53 4 two CARDINAL cana-1295 55 1 two CARDINAL cana-1295 55 2 π‘Ž ∈ 𝐴 ∩ 𝐡 ORG cana-1295 56 1 π‘Ž ∈ 𝐴 ∩ 𝐡 ORG cana-1295 56 2 π‘Ž ∈ 𝐴 ORG cana-1295 56 3 𝑏 ORG cana-1295 56 4 π‘π‘π‘Ž ∈ 𝐴 ORG cana-1295 56 5 𝐴 ∩ 𝐡 ORG cana-1295 56 6 𝑇. NORP cana-1295 56 7 3.5 CARDINAL cana-1295 57 1 ∈ ORG cana-1295 57 2 𝑇 & let 𝐴 ORG cana-1295 57 3 ∈ 𝐴 ORG cana-1295 57 4 𝑏 ORG cana-1295 57 5 π‘Ž ∈ 𝐴 ORG cana-1295 57 6 ∈ ORG cana-1295 57 7 ∈ ORG cana-1295 58 1 ∈ PRODUCT cana-1295 58 2 ∈ ORG cana-1295 58 3 ∈ ORG cana-1295 58 4 ∈ 𝐴. ORG cana-1295 58 5 1074 CARDINAL cana-1295 59 1 7s CARDINAL cana-1295 59 2 2024 CARDINAL cana-1295 59 3 207 CARDINAL cana-1295 59 4 𝑇. NORP cana-1295 59 5 3.6 CARDINAL cana-1295 59 6 two CARDINAL cana-1295 59 7 𝑇. NORP cana-1295 59 8 𝐼1 PRODUCT cana-1295 59 9 two CARDINAL cana-1295 60 1 ∈ ORG cana-1295 60 2 𝐼1 PRODUCT cana-1295 60 3 𝐼1 PRODUCT cana-1295 60 4 𝐼1 βˆͺ MONEY cana-1295 60 5 𝐼1 βˆͺ MONEY cana-1295 60 6 𝐼2 PRODUCT cana-1295 60 7 𝑇. NORP cana-1295 60 8 3.7 CARDINAL cana-1295 60 9 𝑇. NORP cana-1295 61 1 ⋃ DATE cana-1295 61 2 𝐴 ORG cana-1295 61 3 of𝑇. GPE cana-1295 61 4 ∈ 𝐴 ORG cana-1295 61 5 𝑏 ORG cana-1295 61 6 π‘Ž ∈ 𝐴 ORG cana-1295 61 7 ⋃ PRODUCT cana-1295 61 8 ∈ ORG cana-1295 62 1 ∈ PRODUCT cana-1295 62 2 ∈ ORG cana-1295 62 3 π‘π‘π‘Ž ∈ 𝐴. ORG cana-1295 62 4 𝑇. NORP cana-1295 62 5 3.8 CARDINAL cana-1295 62 6 𝐴1 PRODUCT cana-1295 62 7 𝑇1 CARDINAL cana-1295 62 8 𝑇1 PRODUCT cana-1295 62 9 𝐴1 PRODUCT cana-1295 62 10 ∈ ORG cana-1295 62 11 𝐴1 PRODUCT cana-1295 62 12 ∈ ORG cana-1295 62 13 ∈ ORG cana-1295 63 1 3.9 CARDINAL cana-1295 63 2 𝐴 of a bi-ternary semi ORG cana-1295 64 1 𝑇𝐴𝑇 ORG cana-1295 64 2 βŠ† CARDINAL cana-1295 64 3 3.10 CARDINAL cana-1295 64 4 𝑇1 βˆͺ MONEY cana-1295 64 5 ∈ ORG cana-1295 64 6 t. 𝐼 = 𝐼1 PERSON cana-1295 64 7 0 CARDINAL cana-1295 64 8 0 CARDINAL cana-1295 66 1 3.11 CARDINAL cana-1295 66 2 two CARDINAL cana-1295 66 3 𝑇. NORP cana-1295 66 4 𝐴 ORG cana-1295 66 5 two CARDINAL cana-1295 66 6 𝑇. NORP cana-1295 67 1 π‘Ž ∈ 𝐴 ∩ 𝐡 ORG cana-1295 67 2 π‘Ž ∈ 𝐴 ORG cana-1295 67 3 1074 CARDINAL cana-1295 68 1 7s CARDINAL cana-1295 68 2 2024 CARDINAL cana-1295 68 3 208 CARDINAL cana-1295 68 4 π‘Ž ∈ 𝐴 ORG cana-1295 68 5 𝑏 ORG cana-1295 68 6 𝐴 ORG cana-1295 68 7 ∈ 𝐴. PERSON cana-1295 68 8 ∈ ORG cana-1295 68 9 ∈ 𝐴 ORG cana-1295 68 10 ∈ ORG cana-1295 68 11 ∈ 𝐴 ∩ 𝐡. ORG cana-1295 68 12 𝐴 ∩ 𝐡 ORG cana-1295 68 13 𝑇. NORP cana-1295 68 14 3.12 CARDINAL cana-1295 68 15 𝑇. NORP cana-1295 68 16 π‘Ž ∈ 𝐴 ORG cana-1295 68 17 𝑏 ORG cana-1295 68 18 π‘Ž ∈ 𝐴 ORG cana-1295 68 19 ∈ ORG cana-1295 68 20 ∈ ORG cana-1295 69 1 ∈ PRODUCT cana-1295 70 1 ∈ ORG cana-1295 72 1 𝐴 ORG cana-1295 72 2 𝑇. NORP cana-1295 72 3 3.13 CARDINAL cana-1295 72 4 two CARDINAL cana-1295 72 5 𝑇. NORP cana-1295 72 6 𝐼1 CARDINAL cana-1295 72 7 two CARDINAL cana-1295 72 8 ∈ ORG cana-1295 72 9 𝐼1 ⟹ π›Όπ‘Žπ›½ PERCENT cana-1295 72 10 𝐼1 PRODUCT cana-1295 72 11 𝐼1 PRODUCT cana-1295 72 12 𝐼1 βˆͺ MONEY cana-1295 72 13 𝐼1 βˆͺ MONEY cana-1295 72 14 𝐼2 PRODUCT cana-1295 72 15 𝑇. NORP cana-1295 72 16 3.14 CARDINAL cana-1295 72 17 𝑇. NORP cana-1295 72 18 ∈ ORG cana-1295 73 1 ⋃ DATE cana-1295 73 2 𝐴 ORG cana-1295 73 3 𝑇. NORP cana-1295 73 4 π‘Ž ∈ 𝐴 ORG cana-1295 73 5 𝑏 ORG cana-1295 73 6 π‘Ž ∈ 𝐴 ORG cana-1295 73 7 ⋃ PRODUCT cana-1295 73 8 ∈ ORG cana-1295 74 1 ∈ PRODUCT cana-1295 75 1 ∈ ORG cana-1295 75 2 ⋃ PRODUCT cana-1295 77 1 𝐴 ORG cana-1295 77 2 𝑇. NORP cana-1295 77 3 3.15 CARDINAL cana-1295 78 1 𝐴1 PRODUCT cana-1295 78 2 𝑇1 CARDINAL cana-1295 78 3 𝑇1 PRODUCT cana-1295 78 4 𝐴1 PRODUCT cana-1295 78 5 ∈ ORG cana-1295 78 6 𝐴1 PRODUCT cana-1295 78 7 1074 CARDINAL cana-1295 79 1 7s CARDINAL cana-1295 79 2 2024 CARDINAL cana-1295 79 3 209 CARDINAL cana-1295 80 1 3.16 CARDINAL cana-1295 81 1 𝐴𝑇𝑇 βŠ† 𝐴. ORG cana-1295 82 1 3.17:𝑇 CARDINAL cana-1295 82 2 𝑇1 βˆͺ 𝑇2 MONEY cana-1295 82 3 𝑇1 CARDINAL cana-1295 82 4 ∈ ORG cana-1295 82 5 𝑇. 𝐼 ORG cana-1295 82 6 0 CARDINAL cana-1295 83 1 𝑇1& 𝑇2respectively ORG cana-1295 84 1 𝐼is PERSON cana-1295 84 2 𝑇. NORP cana-1295 84 3 3.18 CARDINAL cana-1295 85 1 𝐼1 CARDINAL cana-1295 85 2 𝐼2 CARDINAL cana-1295 85 3 two CARDINAL cana-1295 85 4 𝑇. NORP cana-1295 85 5 𝐼1 ∩ PRODUCT cana-1295 85 6 ∈ ORG cana-1295 85 7 𝐼1 PRODUCT cana-1295 85 8 𝐼1 PRODUCT cana-1295 85 9 𝐼1 ∩ 𝐼2 PRODUCT cana-1295 85 10 𝐼1 ∩ 𝐼2 PRODUCT cana-1295 85 11 3.19 CARDINAL cana-1295 85 12 3.20 CARDINAL cana-1295 85 13 𝐼1 CARDINAL cana-1295 85 14 two CARDINAL cana-1295 86 1 ∈ ORG cana-1295 86 2 𝐼1 PRODUCT cana-1295 86 3 𝐼1 PRODUCT cana-1295 86 4 𝐼1 βˆͺ MONEY cana-1295 86 5 𝐼1 βˆͺ MONEY cana-1295 86 6 𝐼2 PRODUCT cana-1295 86 7 rightideal of𝑇. ORG cana-1295 86 8 3.21 CARDINAL cana-1295 86 9 π‘π‘π‘Ž ∈ 𝐴 ORG cana-1295 86 10 ∈ 𝐴 ORG cana-1295 86 11 3.22 CARDINAL cana-1295 86 12 𝐴2 DATE cana-1295 88 1 𝐴 ORG cana-1295 88 2 1074 CARDINAL cana-1295 89 1 7s CARDINAL cana-1295 89 2 2024 CARDINAL cana-1295 89 3 210 CARDINAL cana-1295 90 1 3.24 CARDINAL cana-1295 90 2 π‘¨πŸ‘ = 𝑨. PERSON cana-1295 90 3 3.25 CARDINAL cana-1295 90 4 3.26 CARDINAL cana-1295 90 5 1 CARDINAL cana-1295 90 6 𝑇. NORP cana-1295 90 7 𝑇. NORP cana-1295 90 8 1 CARDINAL cana-1295 90 9 𝑇. 1 βˆ‰ 𝑀. ORG cana-1295 90 10 𝑇. NORP cana-1295 90 11 𝑇. NORP cana-1295 90 12 𝑇. NORP cana-1295 90 13 𝑇. NORP cana-1295 91 1 𝑀is PERSON cana-1295 91 2 𝑇. NORP cana-1295 91 3 3.27 CARDINAL cana-1295 91 4 𝑇. NORP cana-1295 91 5 𝑇containing 𝐴 ORG cana-1295 91 6 3.28 CARDINAL cana-1295 91 7 a. proof PERSON cana-1295 92 1 π‘†π‘ βˆˆβˆ† CARDINAL cana-1295 93 1 𝐴 ORG cana-1295 93 2 𝐴 βŠ† ORG cana-1295 95 1 𝐴 βŠ† ORG cana-1295 95 2 ∈ ORG cana-1295 95 3 𝑇containing 𝐴. PERSON cana-1295 95 4 3.34 CARDINAL cana-1295 95 5 𝑄1 & 𝑄2 ORG cana-1295 95 6 𝑇1& 𝑇2 ORG cana-1295 95 7 𝑇1 ∩ PRODUCT cana-1295 95 8 𝑇1 CARDINAL cana-1295 95 9 𝑇1 CARDINAL cana-1295 95 10 βŠ† CARDINAL cana-1295 95 11 𝑄1 ORG cana-1295 95 12 𝑄2𝑇2 𝑇2 DATE cana-1295 95 13 𝑇2𝑄2𝑇2 GPE cana-1295 97 1 𝑇. NORP cana-1295 97 2 3.35 CARDINAL cana-1295 97 3 𝑃2 PERSON cana-1295 97 4 𝑃1 & 𝑃2 ORG cana-1295 97 5 𝑋1 LOC cana-1295 97 6 π‘Œ2 CARDINAL cana-1295 98 1 βŠ† CARDINAL cana-1295 98 2 βŠ† ORG cana-1295 98 3 𝑍2 βŠ† ORG cana-1295 98 4 𝑃2 PRODUCT cana-1295 98 5 3.36 CARDINAL cana-1295 98 6 𝑃2 PERSON cana-1295 98 7 𝑃1 & 𝑃2 ORG cana-1295 98 8 ∈ 𝑃1π‘œπ‘Ÿπ‘¦1 ∈ ORG cana-1295 98 9 ∈ ORG cana-1295 98 10 𝑃1 π‘₯2𝑦2𝑧2 MONEY cana-1295 98 11 ∈ ORG cana-1295 98 12 𝑃2 ⟹ PERSON cana-1295 98 13 ∈ ORG cana-1295 98 14 ∈ ORG cana-1295 98 15 𝑃2 PRODUCT cana-1295 98 16 1074 CARDINAL cana-1295 99 1 7s CARDINAL cana-1295 99 2 2024 CARDINAL cana-1295 99 3 211 CARDINAL cana-1295 99 4 3.37 CARDINAL cana-1295 101 1 𝑇. let 𝐴 ORG cana-1295 101 2 𝐴 ORG cana-1295 103 1 𝐴𝐡𝐢 βŠ† 𝑃 ORG cana-1295 103 2 βŠ† CARDINAL cana-1295 103 3 𝑇. NORP cana-1295 103 4 4.1 CARDINAL cana-1295 104 1 𝑇1,∘ NORP cana-1295 104 2 two CARDINAL cana-1295 106 1 𝑆2to GPE cana-1295 106 2 1 CARDINAL cana-1295 107 1 βˆ…(π‘Ž PERSON cana-1295 107 2 ∘ CARDINAL cana-1295 107 3 ∘ CARDINAL cana-1295 107 4 2 CARDINAL cana-1295 107 5 ∈ 𝑆 & βˆ…(π‘Ž ORG cana-1295 107 6 4.2 CARDINAL cana-1295 107 7 𝑇. NORP cana-1295 107 8 4.3 CARDINAL cana-1295 107 9 one CARDINAL cana-1295 107 10 𝑇. NORP cana-1295 107 11 4.4 CARDINAL cana-1295 108 1 one CARDINAL cana-1295 108 2 𝑆 ORG cana-1295 108 3 𝑇. 𝑆 β‰… ORG cana-1295 108 4 𝑇. NORP cana-1295 108 5 4.5 CARDINAL cana-1295 109 1 4.6 CARDINAL cana-1295 109 2 𝑒′ ∈ 𝑇 PERSON cana-1295 109 3 𝑇. NORP cana-1295 109 4 4.7 CARDINAL cana-1295 109 5 𝑆2 ORG cana-1295 109 6 𝑇1 βˆͺ 𝑇2 MONEY cana-1295 109 7 two CARDINAL cana-1295 110 1 𝑆 ORG cana-1295 111 1 ∈ ORG cana-1295 111 2 βˆ…(π‘Ž PERSON cana-1295 111 3 0β€² CARDINAL cana-1295 111 4 first ORDINAL cana-1295 112 1 0 CARDINAL cana-1295 112 2 0(zero CARDINAL cana-1295 112 3 0β€² CARDINAL cana-1295 112 4 0(zero CARDINAL cana-1295 112 5 𝑇. NORP cana-1295 113 1 𝑆2 β‡’ βˆ…1(0 GPE cana-1295 113 2 0β€² CARDINAL cana-1295 114 1 0 CARDINAL cana-1295 115 1 1074 CARDINAL cana-1295 116 1 7s CARDINAL cana-1295 116 2 2024 CARDINAL cana-1295 116 3 212 CARDINAL cana-1295 117 1 ∈ ORG cana-1295 117 2 ∈ ORG cana-1295 118 1 0β€² CARDINAL cana-1295 119 1 𝑏 ∈ π‘˜π‘’π‘Ÿβˆ… & 𝑠 ORG cana-1295 119 2 0β€²& βˆ…(b ORG cana-1295 119 3 0β€² CARDINAL cana-1295 120 1 0β€². CARDINAL cana-1295 120 2 0β€². βˆ…(𝑠 GPE cana-1295 120 3 = βˆ…(𝑠 PERSON cana-1295 120 4 0β€² CARDINAL cana-1295 121 1 π‘Žπ‘π‘  ∈ π‘˜π‘’π‘Ÿβˆ… PERSON cana-1295 121 2 ∈ ORG cana-1295 121 3 ∈ ORG cana-1295 121 4 𝑏 ∈ π‘˜π‘’π‘Ÿβˆ… &𝑠 ORG cana-1295 121 5 ∈ ORG cana-1295 122 1 s. theorem 4.8 ORG cana-1295 123 1 𝐴 ORG cana-1295 123 2 𝑆 ORG cana-1295 123 3 𝑇. NORP cana-1295 123 4 0 & 0β€² ORG cana-1295 124 1 𝑆 & 𝑇 ORG cana-1295 125 1 ∈ PRODUCT cana-1295 125 2 βˆƒπœ‡ ∈ PERSON cana-1295 126 1 0β€² CARDINAL cana-1295 128 1 βŠ† CARDINAL cana-1295 128 2 ∈ PRODUCT cana-1295 128 3 𝑏 ∈ 𝐴 ORG cana-1295 128 4 π‘Žβ€² ORG cana-1295 128 5 𝑇. NORP cana-1295 128 6 π‘Žβ€²π‘β€²π‘‘ ∈ ORG cana-1295 128 7 π‘Žβ€²π‘β€²π‘‘ = βˆ…(π‘Ž)βˆ…(𝑏)βˆ…(𝑠 ORG cana-1295 128 8 𝑏 ∈ 𝐴 &𝑠 ORG cana-1295 128 9 s. ∴ π‘Žβ€²π‘β€²π‘‘ ∈ ORG cana-1295 128 10 ∈ ORG cana-1295 128 11 ∈ ORG cana-1295 128 12 𝑇. hereafter βˆ…(𝐴 ORG cana-1295 128 13 𝑇. NORP cana-1295 128 14 5.1 CARDINAL cana-1295 128 15 𝑇 ORG cana-1295 128 16 𝑇1 CARDINAL cana-1295 128 17 & 𝑇2 DATE cana-1295 129 1 𝑇1 & 𝑇2 ORG cana-1295 129 2 non zero(0 ORG cana-1295 129 3 multiplicative ORG cana-1295 130 1 1074 CARDINAL cana-1295 131 1 7s CARDINAL cana-1295 131 2 2024 CARDINAL cana-1295 131 3 213 CARDINAL cana-1295 131 4 1 CARDINAL cana-1295 131 5 rao NORP cana-1295 131 6 d. m. PERSON cana-1295 131 7 rao ORG cana-1295 131 8 2014 DATE cana-1295 133 1 1(7 CARDINAL cana-1295 133 2 105 CARDINAL cana-1295 134 1 2 CARDINAL cana-1295 134 2 rao ORG cana-1295 134 3 d. m. PERSON cana-1295 134 4 rao ORG cana-1295 134 5 2015 CARDINAL cana-1295 136 1 3 CARDINAL cana-1295 136 2 49 CARDINAL cana-1295 137 1 3 CARDINAL cana-1295 137 2 a., & rao ORG cana-1295 137 3 d. m. PERSON cana-1295 137 4 2013 DATE cana-1295 139 1 76 DATE cana-1295 139 2 848 CARDINAL cana-1295 140 1 4 CARDINAL cana-1295 140 2 dixit ORG cana-1295 140 3 n., & dewan ORG cana-1295 140 4 s. (1995 ORG cana-1295 142 1 18 CARDINAL cana-1295 142 2 501 CARDINAL cana-1295 143 1 kar ORG cana-1295 143 2 s., & maity ORG cana-1295 143 3 b. k. ( PERSON cana-1295 143 4 2011 DATE cana-1295 145 1 the alexandru ioan cuza university-mathematics ORG cana-1295 145 2 57(2 CARDINAL cana-1295 145 3 247 CARDINAL cana-1295 146 1 5 CARDINAL cana-1295 146 2 r. NORP cana-1295 146 3 baupradist GPE cana-1295 146 4 s., & saelee ORG cana-1295 146 5 s. PERSON cana-1295 146 6 2012 DATE cana-1295 149 1 j. phys PERSON cana-1295 150 1 sci ORG cana-1295 150 2 7 DATE cana-1295 150 3 2674 CARDINAL cana-1295 151 1 6 CARDINAL cana-1295 151 2 r., & pawar ORG cana-1295 151 3 y. s. PERSON cana-1295 151 4 2012 DATE cana-1295 153 1 kyungpook mathematical journal ORG cana-1295 153 2 52(4 CARDINAL cana-1295 154 1 7 CARDINAL cana-1295 154 2 a., & rao ORG cana-1295 154 3 m. d. PERSON cana-1295 154 4 2019 DATE cana-1295 155 1 8(6 CARDINAL cana-1295 155 2 4 CARDINAL cana-1295 155 3 1403-1404 DATE cana-1295 156 1 8 CARDINAL cana-1295 156 2 a. PERSON cana-1295 156 3 2007 DATE cana-1295 158 1 ΡƒΠΊΡ€Π°Ρ—Π½ΡΡŒΠΊΠΈΠΉ ΠΌΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π½ΠΈΠΉ вісник PERSON cana-1295 159 1 9 CARDINAL cana-1295 159 2 r. NORP cana-1295 159 3 2012 DATE cana-1295 162 1 j. pure PERSON cana-1295 162 2 639 CARDINAL cana-1295 163 1 10 CARDINAL cana-1295 163 2 vasantha kandasamy PERSON cana-1295 163 3 w.b. GPE cana-1295 163 4 bialgebraic PERSON cana-1295 163 5 american research press ORG cana-1295 163 6 rehoboth GPE cana-1295 163 7 2003 DATE cana-1295 164 1 11 CARDINAL cana-1295 164 2 vasantha kandasamy PERSON cana-1295 164 3 w.b. GPE cana-1295 164 4 sci ORG cana-1295 164 5 phy ORG cana-1295 165 1 sci. ORG cana-1295 165 2 11 CARDINAL cana-1295 165 3 1999 DATE cana-1295 165 4 186 CARDINAL cana-1295 166 1 12 CARDINAL cana-1295 166 2 g. srinivasa rao PERSON cana-1295 166 3 mathematics & mathematical sciences ORG cana-1295 166 4 9(2 CARDINAL cana-1295 166 5 2016 CARDINAL cana-1295 166 6 185 CARDINAL cana-1295 167 1 13 CARDINAL cana-1295 167 2 d. madhusudhana rao PERSON cana-1295 167 3 g. srinivasa rao PERSON cana-1295 167 4 4(11 DATE cana-1295 167 5 2014 DATE cana-1295 167 6 123-130 CARDINAL cana-1295 168 1 14 CARDINAL cana-1295 168 2 g. srinivasa rao PERSON cana-1295 168 3 d. madhusudhana rao PERSON cana-1295 169 1 j. PERSON cana-1295 170 1 5(12 CARDINAL cana-1295 170 2 2014 DATE cana-1295 170 3 24-30 DATE cana-1295 171 1 15 CARDINAL cana-1295 171 2 g. srinivasa rao PERSON cana-1295 171 3 d. madhusudhana rao PERSON cana-1295 172 1 2(1 CARDINAL cana-1295 172 2 2015 CARDINAL cana-1295 172 3 3 CARDINAL cana-1295 173 1 16 CARDINAL cana-1295 173 2 g. srinivasa rao PERSON cana-1295 173 3 a. nagamalleswara rao PERSON cana-1295 173 4 varma PERSON cana-1295 173 5 d.madhusudhana rao PERSON cana-1295 173 6 tgsr GPE cana-1295 173 7 malaya GPE cana-1295 173 8 no.1 ORG cana-1295 173 9 546 CARDINAL cana-1295 173 10 2021 DATE cana-1295 174 1 17 CARDINAL cana-1295 174 2 g. srinivasa rao PERSON cana-1295 174 3 a. nagamalleswara rao PERSON cana-1295 174 4 varma PERSON cana-1295 174 5 d. madhusudhana rao PERSON cana-1295 174 6 10 CARDINAL cana-1295 174 7 2021 CARDINAL cana-1295 174 8 no.3 GPE cana-1295 174 9 1183-1195 CARDINAL