{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T23:07:48Z","timestamp":1783897668306,"version":"3.55.0"},"reference-count":8,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article we formalize in Mizar [1], [2] a derivation of commutative rings, its definition and some properties. The details are to be referred to [5], [7]. A derivation of a ring, say\n                    <jats:italic>D<\/jats:italic>\n                    , is defined generally as a map from a commutative ring\n                    <jats:italic>A<\/jats:italic>\n                    to\n                    <jats:italic>A<\/jats:italic>\n                    -Module\n                    <jats:italic>M<\/jats:italic>\n                    with specific conditions. However we start with simpler case, namely dom\n                    <jats:italic>D<\/jats:italic>\n                    = rng\n                    <jats:italic>D<\/jats:italic>\n                    . This allows to define a derivation in other rings such as a polynomial ring.\n                  <\/jats:p>\n                  <jats:p>\n                    A derivation is a map\n                    <jats:italic>D<\/jats:italic>\n                    :\n                    <jats:italic>A \u2192 A<\/jats:italic>\n                    satisfying the following conditions:\n                  <\/jats:p>\n                  <jats:p>\n                    (i)\n                    <jats:italic>D<\/jats:italic>\n                    (\n                    <jats:italic>x<\/jats:italic>\n                    +\n                    <jats:italic>y<\/jats:italic>\n                    ) =\n                    <jats:italic>Dx<\/jats:italic>\n                    +\n                    <jats:italic>Dy<\/jats:italic>\n                    ,\n                  <\/jats:p>\n                  <jats:p>\n                    (ii)\n                    <jats:italic>D<\/jats:italic>\n                    (\n                    <jats:italic>xy<\/jats:italic>\n                    ) =\n                    <jats:italic>xDy<\/jats:italic>\n                    +\n                    <jats:italic>yDx<\/jats:italic>\n                    , \u2200\n                    <jats:italic>x, y<\/jats:italic>\n                    \u2208\n                    <jats:italic>A<\/jats:italic>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    Typical properties are formalized such as:\n                    <jats:disp-formula>\n                      <jats:alternatives>\n                        <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forma-2021-0001_eq_001.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n                          <m:mrow>\n                            <m:mi>D<\/m:mi>\n                            <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                <m:munderover>\n                                  <m:mo>\u2211<\/m:mo>\n                                  <m:mrow>\n                                    <m:mi>i<\/m:mi>\n                                    <m:mo>=<\/m:mo>\n                                    <m:mn>1<\/m:mn>\n                                  <\/m:mrow>\n                                  <m:mi>n<\/m:mi>\n                                <\/m:munderover>\n                                <m:mrow>\n                                  <m:msub>\n                                    <m:mrow>\n                                      <m:mi>x<\/m:mi>\n                                    <\/m:mrow>\n                                    <m:mi>i<\/m:mi>\n                                  <\/m:msub>\n                                <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                            <\/m:mrow>\n                            <m:mo>=<\/m:mo>\n                            <m:munderover>\n                              <m:mo>\u2211<\/m:mo>\n                              <m:mrow>\n                                <m:mi>i<\/m:mi>\n                                <m:mo>=<\/m:mo>\n                                <m:mn>1<\/m:mn>\n                              <\/m:mrow>\n                              <m:mi>n<\/m:mi>\n                            <\/m:munderover>\n                            <m:mrow>\n                              <m:mi>D<\/m:mi>\n                              <m:msub>\n                                <m:mrow>\n                                  <m:mi>x<\/m:mi>\n                                <\/m:mrow>\n                                <m:mi>i<\/m:mi>\n                              <\/m:msub>\n                            <\/m:mrow>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>D\\left( {\\sum\\limits_{i = 1}^n {{x_i}} } \\right) = \\sum\\limits_{i = 1}^n {D{x_i}}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:disp-formula>\n                    and\n                    <jats:disp-formula>\n                      <jats:alternatives>\n                        <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forma-2021-0001_eq_002.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n                          <m:mrow>\n                            <m:mi>D<\/m:mi>\n                            <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                <m:munderover>\n                                  <m:mo>\u220f<\/m:mo>\n                                  <m:mrow>\n                                    <m:mi>i<\/m:mi>\n                                    <m:mo>=<\/m:mo>\n                                    <m:mn>1<\/m:mn>\n                                  <\/m:mrow>\n                                  <m:mi>n<\/m:mi>\n                                <\/m:munderover>\n                                <m:mrow>\n                                  <m:msub>\n                                    <m:mrow>\n                                      <m:mi>x<\/m:mi>\n                                    <\/m:mrow>\n                                    <m:mi>i<\/m:mi>\n                                  <\/m:msub>\n                                <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                            <\/m:mrow>\n                            <m:mo>=<\/m:mo>\n                            <m:munderover>\n                              <m:mo>\u2211<\/m:mo>\n                              <m:mrow>\n                                <m:mi>i<\/m:mi>\n                                <m:mo>=<\/m:mo>\n                                <m:mn>1<\/m:mn>\n                              <\/m:mrow>\n                              <m:mi>n<\/m:mi>\n                            <\/m:munderover>\n                            <m:mrow>\n                              <m:msub>\n                                <m:mrow>\n                                  <m:mi>x<\/m:mi>\n                                <\/m:mrow>\n                                <m:mn>1<\/m:mn>\n                              <\/m:msub>\n                              <m:msub>\n                                <m:mrow>\n                                  <m:mi>x<\/m:mi>\n                                <\/m:mrow>\n                                <m:mn>2<\/m:mn>\n                              <\/m:msub>\n                              <m:mo>\u22ef<\/m:mo>\n                              <m:mi>D<\/m:mi>\n                              <m:msub>\n                                <m:mrow>\n                                  <m:mi>x<\/m:mi>\n                                <\/m:mrow>\n                                <m:mi>i<\/m:mi>\n                              <\/m:msub>\n                              <m:mo>\u22ef<\/m:mo>\n                              <m:msub>\n                                <m:mrow>\n                                  <m:mi>x<\/m:mi>\n                                <\/m:mrow>\n                                <m:mi>n<\/m:mi>\n                              <\/m:msub>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                <m:mo>\u2200<\/m:mo>\n                                <m:msub>\n                                  <m:mrow>\n                                    <m:mi>x<\/m:mi>\n                                  <\/m:mrow>\n                                  <m:mi>i<\/m:mi>\n                                <\/m:msub>\n                                <m:mo>\u2208<\/m:mo>\n                                <m:mi>A<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                            <\/m:mrow>\n                            <m:mo>.<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>D\\left( {\\prod\\limits_{i = 1}^n {{x_i}} } \\right) = \\sum\\limits_{i = 1}^n {{x_1}{x_2} \\cdots D{x_i} \\cdots {x_n}} \\left( {\\forall {x_i} \\in A} \\right).<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:disp-formula>\n                  <\/jats:p>\n                  <jats:p>\n                    We also formalized the Leibniz Formula for power of derivation\n                    <jats:italic>D<\/jats:italic>\n                    :\n                    <jats:disp-formula>\n                      <jats:alternatives>\n                        <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forma-2021-0001_eq_003.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n                          <m:mrow>\n                            <m:msup>\n                              <m:mrow>\n                                <m:mi>D<\/m:mi>\n                              <\/m:mrow>\n                              <m:mi>n<\/m:mi>\n                            <\/m:msup>\n                            <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                <m:mi>x<\/m:mi>\n                                <m:mi>y<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                            <\/m:mrow>\n                            <m:mo>=<\/m:mo>\n                            <m:munderover>\n                              <m:mo>\u2211<\/m:mo>\n                              <m:mrow>\n                                <m:mi>i<\/m:mi>\n                                <m:mo>=<\/m:mo>\n                                <m:mn>0<\/m:mn>\n                              <\/m:mrow>\n                              <m:mi>n<\/m:mi>\n                            <\/m:munderover>\n                            <m:mrow>\n                              <m:mrow>\n                                <m:mo>(<\/m:mo>\n                                <m:mrow>\n                                  <m:msubsup>\n                                    <m:mrow\/>\n                                    <m:mi>i<\/m:mi>\n                                    <m:mi>n<\/m:mi>\n                                  <\/m:msubsup>\n                                <\/m:mrow>\n                                <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:msup>\n                                <m:mrow>\n                                  <m:mi>D<\/m:mi>\n                                <\/m:mrow>\n                                <m:mi>i<\/m:mi>\n                              <\/m:msup>\n                              <m:mi>x<\/m:mi>\n                              <m:msup>\n                                <m:mrow>\n                                  <m:mi>D<\/m:mi>\n                                <\/m:mrow>\n                                <m:mrow>\n                                  <m:mi>n<\/m:mi>\n                                  <m:mo>-<\/m:mo>\n                                  <m:mi>i<\/m:mi>\n                                <\/m:mrow>\n                              <\/m:msup>\n                              <m:mi>y<\/m:mi>\n                              <m:mo>.<\/m:mo>\n                            <\/m:mrow>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>{D^n}\\left( {xy} \\right) = \\sum\\limits_{i = 0}^n {\\left( {_i^n} \\right){D^i}x{D^{n - i}}y.}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:disp-formula>\n                  <\/jats:p>\n                  <jats:p>\n                    Lastly applying the definition to the polynomial ring of\n                    <jats:italic>A<\/jats:italic>\n                    and a derivation of polynomial ring was formalized. We mentioned a justification about compatibility of the derivation in this article to the same object that has treated as differentiations of polynomial functions [3].\n                  <\/jats:p>","DOI":"10.2478\/forma-2021-0001","type":"journal-article","created":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T11:48:37Z","timestamp":1630064917000},"page":"1-8","source":"Crossref","is-referenced-by-count":3,"title":["Derivation of Commutative Rings and the Leibniz Formula for Power of Derivation"],"prefix":"10.2478","volume":"29","author":[{"given":"Yasushige","family":"Watase","sequence":"first","affiliation":[{"name":"Suginami-ku Matsunoki , 3-21-6 Tokyo , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2021,8,26]]},"reference":[{"key":"2026071212491089526_j_forma-2021-0001_ref_001","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.10.1007\/978-3-319-20615-8_17","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026071212491089526_j_forma-2021-0001_ref_002","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026071212491089526_j_forma-2021-0001_ref_003","doi-asserted-by":"crossref","unstructured":"[3] Artur Korni\u0142owicz. Differentiability of polynomials over reals. Formalized Mathematics, 25(1):31\u201337, 2017. doi:10.1515\/forma-2017-0002.10.1515\/forma-2017-0002","DOI":"10.1515\/forma-2017-0002"},{"key":"2026071212491089526_j_forma-2021-0001_ref_004","doi-asserted-by":"crossref","unstructured":"[4] Artur Korni\u0142owicz and Christoph Schwarzweller. The first isomorphism theorem and other properties of rings. Formalized Mathematics, 22(4):291\u2013301, 2014. doi:10.2478\/forma-2014-0029.10.2478\/forma-2014-0029","DOI":"10.2478\/forma-2014-0029"},{"key":"2026071212491089526_j_forma-2021-0001_ref_005","unstructured":"[5] Hideyuki Matsumura. Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2nd edition, 1989."},{"key":"2026071212491089526_j_forma-2021-0001_ref_006","unstructured":"[6] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461\u2013470, 2001."},{"key":"2026071212491089526_j_forma-2021-0001_ref_007","unstructured":"[7] Masayoshi Nagata. Theory of Commutative Fields, volume 125 of Translations of Mathematical Monographs. American Mathematical Society, 1985."},{"key":"2026071212491089526_j_forma-2021-0001_ref_008","doi-asserted-by":"crossref","unstructured":"[8] Christoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185\u2013195, 2017. doi:10.1515\/forma-2017-0018.10.1515\/forma-2017-0018","DOI":"10.1515\/forma-2017-0018"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2021-0001","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T22:56:51Z","timestamp":1783897011000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2021-0001"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,1]]},"references-count":8,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,8,26]]},"published-print":{"date-parts":[[2021,4,1]]}},"alternative-id":["10.2478\/forma-2021-0001"],"URL":"https:\/\/doi.org\/10.2478\/forma-2021-0001","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,1]]}}}