{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:f05f6f83-0600-4f4e-951c-348c86414c39",
  "labels": {
    "zh-CN": "环（不要求交换）",
    "en": "ring"
  },
  "alternative_labels": [
    {
      "lang": "zh-CN",
      "value": "环"
    }
  ],
  "scope_note": "§28.1 explicitly allows noncommutative multiplication.",
  "identity_status": "active",
  "identity_revision": "a7155ad40676ce975cac08fc434cf11aa23fa21109461668c4b1280c23ce15cb",
  "occurrences": [
    {
      "concept_id": "urn:uuid:f05f6f83-0600-4f4e-951c-348c86414c39",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "zh-CN",
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "anchor": "first-bb6b914065450a4d0c14-zh",
      "chapter": "28",
      "section": "28.1",
      "offset": 221,
      "range": [
        7830,
        8103
      ],
      "source_revision": "d2a329a1c341b3d909aed292f756d80691f87ff3253e09de676b6b167a5fefb8",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "环(ring)",
      "context": {
        "range": [
          7823,
          8618
        ],
        "quote": "在数学中，“[<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/ring\" data-concept-identity=\"urn:uuid:f05f6f83-0600-4f4e-951c-348c86414c39\" data-concept-route=\"/concept-identities#concept-f05f6f83-0600-4f4e-951c-348c86414c39\" id=\"first-bb6b914065450a4d0c14-zh\">环(ring)</u>](../../../../../concepts/algebra/ring)”是带加法和乘法的集合，这些运算满足结合律与分配律，但不一定满足交换律；“[<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/algebra-over-a-field\" data-concept-identity=\"urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4\" data-concept-route=\"/concept-identities#concept-7e2e6da6-b92e-45a5-b490-c11370b027b4\" id=\"first-33ec330b5f5555922192-zh\">代数(algebra)</u>](../../../../../concepts/algebra/algebra-over-a-field)”则是同时具有某个标量域上向量空间结构的环。正则对易关系与正则反对易关系分别定义了Weyl代数和Clifford代数。以复数为标量的情形最简单，我们先讨论复数情形，再讨论实数情形。",
        "quote_sha256": "06ca4caca9fac91edc206d023f1453a80ed150b88b9c0aa7f6e619ded5ac88a4",
        "environment": null,
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
          "lang": "en",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-22a1667533fa5569b700a77b"
    },
    {
      "concept_id": "urn:uuid:f05f6f83-0600-4f4e-951c-348c86414c39",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "en",
      "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "anchor": "first-bb6b914065450a4d0c14-en",
      "chapter": "28",
      "section": "28.1",
      "offset": 254,
      "range": [
        6964,
        7234
      ],
      "source_revision": "1437e8e29fe07ba5486788969d959b4b958fe09241e74dc8c02204e7032cf358",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "ring",
      "context": {
        "range": [
          6944,
          7964
        ],
        "quote": "In mathematics, a “[<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/ring\" data-concept-identity=\"urn:uuid:f05f6f83-0600-4f4e-951c-348c86414c39\" data-concept-route=\"/concept-identities#concept-f05f6f83-0600-4f4e-951c-348c86414c39\" id=\"first-bb6b914065450a4d0c14-en\">ring</u>](../../../../../concepts/algebra/ring)” is a set with associative and distributive addition and multiplication laws, not necessarily commutative. An “[<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/algebra-over-a-field\" data-concept-identity=\"urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4\" data-concept-route=\"/concept-identities#concept-7e2e6da6-b92e-45a5-b490-c11370b027b4\" id=\"first-33ec330b5f5555922192-en\">algebra</u>](../../../../../concepts/algebra/algebra-over-a-field)” is a ring that is also a vector space over a field of scalars. The canonical commutation and anticommutation relations define the Weyl and Clifford algebras, respectively. We begin with complex scalars and then turn to real scalars.",
        "quote_sha256": "74fdb3eaf823debba65c9941539c3cbd398d959ab7689cf89ff47cdaee6a3b91",
        "environment": null,
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
          "lang": "zh-CN",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-ae97c8fb8799481f013db2ba"
    }
  ],
  "registered_positions": [
    {
      "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "lang": "en",
      "section": "28.1",
      "signal": "term-mark",
      "label": "ring",
      "english": "ring",
      "offset": 254
    },
    {
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "lang": "zh-CN",
      "section": "28.1",
      "signal": "term-mark",
      "label": "环",
      "english": "ring",
      "offset": 221
    }
  ],
  "statements": [],
  "resolved_references": [],
  "reading_order": [
    "occ-22a1667533fa5569b700a77b",
    "occ-ae97c8fb8799481f013db2ba"
  ],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 2,
    "located_occurrences": 2,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "base_revision": "c2422d0e409030519feff9387191988b435a817d740b496141a5f5c057ff9e50",
  "optional_enrichments": []
}