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<!DOCTYPE html><html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1.0"><meta name="generator" content="rustdoc"><meta name="description" content="API documentation for the Rust `AbstractRing` trait in crate `alga`."><meta name="keywords" content="rust, rustlang, rust-lang, AbstractRing"><title>alga::general::AbstractRing - Rust</title><link rel="stylesheet" type="text/css" href="../../normalize.css"><link rel="stylesheet" type="text/css" href="../../rustdoc.css" id="mainThemeStyle"><link rel="stylesheet" type="text/css" href="../../dark.css"><link rel="stylesheet" type="text/css" href="../../light.css" id="themeStyle"><script src="../../storage.js"></script><noscript><link rel="stylesheet" href="../../noscript.css"></noscript><link rel="shortcut icon" href="../../favicon.ico"><style type="text/css">#crate-search{background-image:url("../../down-arrow.svg");}</style></head><body class="rustdoc trait"><!--[if lte IE 8]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="sidebar"><div class="sidebar-menu">☰</div><a href='../../alga/index.html'><div class='logo-container'><img src='../../rust-logo.png' alt='logo'></div></a><p class='location'>Trait AbstractRing</p><div class="sidebar-elems"><div class="block items"><a class="sidebar-title" href="#provided-methods">Provided Methods</a><div class="sidebar-links"><a href="#method.prop_mul_and_add_are_distributive">prop_mul_and_add_are_distributive</a><a href="#method.prop_mul_and_add_are_distributive_approx">prop_mul_and_add_are_distributive_approx</a></div><a class="sidebar-title" href="#foreign-impls">Implementations on Foreign Types</a><div class="sidebar-links"><a href="#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-Complex%3CN%3E">Complex<N></a><a href="#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-f32">f32</a><a href="#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-f64">f64</a><a href="#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i16">i16</a><a href="#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i32">i32</a><a href="#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i64">i64</a><a href="#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i8">i8</a><a href="#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-isize">isize</a></div><a class="sidebar-title" href="#implementors">Implementors</a></div><p class='location'><a href='../index.html'>alga</a>::<wbr><a href='index.html'>general</a></p><script>window.sidebarCurrent = {name: 'AbstractRing', ty: 'trait', relpath: ''};</script><script defer src="sidebar-items.js"></script></div></nav><div class="theme-picker"><button id="theme-picker" aria-label="Pick another theme!"><img src="../../brush.svg" width="18" alt="Pick another theme!"></button><div id="theme-choices"></div></div><script src="../../theme.js"></script><nav class="sub"><form class="search-form js-only"><div class="search-container"><div><select id="crate-search"><option value="All crates">All crates</option></select><input class="search-input" name="search" autocomplete="off" spellcheck="false" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"></div><a id="settings-menu" href="../../settings.html"><img src="../../wheel.svg" width="18" alt="Change settings"></a></div></form></nav><section id="main" class="content"><h1 class='fqn'><span class='out-of-band'><span id='render-detail'><a id="toggle-all-docs" href="javascript:void(0)" title="collapse all docs">[<span class='inner'>−</span>]</a></span><a class='srclink' href='../../src/alga/general/two_operators.rs.html#30-66' title='goto source code'>[src]</a></span><span class='in-band'>Trait <a href='../index.html'>alga</a>::<wbr><a href='index.html'>general</a>::<wbr><a class="trait" href=''>AbstractRing</a></span></h1><div class="docblock type-decl hidden-by-usual-hider"><pre class='rust trait'>pub trait AbstractRing<A: <a class="trait" href="../../alga
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fn <a href='#method.prop_mul_and_add_are_distributive_approx' class='fnname'>prop_mul_and_add_are_distributive_approx</a>(<br> args: <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.tuple.html">(</a>Self, Self, Self<a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.tuple.html">)</a><br> ) -> <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.bool.html">bool</a><br> <span class="where">where<br> Self: <a class="trait" href="../../approx/relative_eq/trait.RelativeEq.html" title="trait approx::relative_eq::RelativeEq">RelativeEq</a></span>,
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{ ... }
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<div class='item-spacer'></div> fn <a href='#method.prop_mul_and_add_are_distributive' class='fnname'>prop_mul_and_add_are_distributive</a>(args: <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.tuple.html">(</a>Self, Self, Self<a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.tuple.html">)</a>) -> <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.bool.html">bool</a><br> <span class="where">where<br> Self: <a class="trait" href="https://doc.rust-lang.org/nightly/core/cmp/trait.Eq.html" title="trait core::cmp::Eq">Eq</a></span>,
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{ ... }
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}</pre></div><div class='docblock'><p>A <strong>ring</strong> is the combination of an Abelian group and a multiplicative monoid structure.</p>
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<p>A ring is equipped with:</p>
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<ul>
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<li>
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<p>An abstract operator (usually the addition, "+") that fulfills the constraints of an Abelian group.</p>
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<p><em>An Abelian group is a set with a closed commutative and associative addition with the divisibility property and an identity element.</em></p>
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</li>
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<li>
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<p>A second abstract operator (usually the multiplication, "×") that fulfills the constraints of a monoid.</p>
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<p><em>A set equipped with a closed associative multiplication with the divisibility property and an identity element.</em></p>
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</li>
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</ul>
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<p>The multiplication is distributive over the addition:</p>
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<h1 id="distributivity" class="section-header"><a href="#distributivity">Distributivity</a></h1>
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<pre><code class="language-notrust">a, b, c ∈ Self, a × (b + c) = a × b + a × c.
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</code></pre>
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</div>
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<h2 id='provided-methods' class='small-section-header'>Provided methods<a href='#provided-methods' class='anchor'></a></h2><div class='methods'><h3 id='method.prop_mul_and_add_are_distributive_approx' class='method'><code id='prop_mul_and_add_are_distributive_approx.v'>fn <a href='#method.prop_mul_and_add_are_distributive_approx' class='fnname'>prop_mul_and_add_are_distributive_approx</a>(args: <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.tuple.html">(</a>Self, Self, Self<a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.tuple.html">)</a>) -> <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.bool.html">bool</a> <span class="where fmt-newline">where<br> Self: <a class="trait" href="../../approx/relative_eq/trait.RelativeEq.html" title="trait approx::relative_eq::RelativeEq">RelativeEq</a>, </span></code></h3><div class='docblock'><p>Returns <code>true</code> if the multiplication and addition operators are distributive for
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the given argument tuple. Approximate equality is used for verifications.</p>
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</div><h3 id='method.prop_mul_and_add_are_distributive' class='method'><code id='prop_mul_and_add_are_distributive.v'>fn <a href='#method.prop_mul_and_add_are_distributive' class='fnname'>prop_mul_and_add_are_distributive</a>(args: <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.tuple.html">(</a>Self, Self, Self<a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.tuple.html">)</a>) -> <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.bool.html">bool</a> <span class="where fmt-newline">where<br> Self: <a class="trait" href="https://doc.rust-lang.org/nightly/core/cmp/trait.Eq.html" title="trait core::cmp::Eq">Eq</a>, </span></code></h3><div class='docblock'><p>Returns <code>true</code> if the multiplication and addition operators are distributive for
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the given argument tuple.</p>
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</div></div><span class='loading-content'>Loading content...</span>
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<h2 id='foreign-impls' class='small-section-header'>Implementations on Foreign Types<a href='#foreign-impls' class='anchor'></a></h2><h3 id='impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i8' class='impl'><code class='in-band'>impl <a class="trait" href="../../alga/general/trait.AbstractRing.html" title="trait alga::general::AbstractRing">AbstractRing</a><<a class="struct" href="../../alga/general/struct.Additive.html" title="struct alga::general::Additive">Additive</a>, <a class="struct" href="../../alga/general/struct.Multiplicative.html" title="struct alga::general::Multiplicative">Multiplicative</a>> for <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.i8.html">i8</a></code><a href='#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i8' class='anchor'></a><a class='srclink' href='../../src/alga/macros.rs.html#102' title='goto source code'>[src]</a></h3><div class='impl-items'></div><h3 id='impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i16' class='impl'><code class='in-band'>impl <a class="trait" href="../../alga/general/trait.AbstractRing.html" title="trait alga::general::AbstractRing">AbstractRing</a><<a class="struct" href="../../alga/general/struct.Additive.html" title="struct alga::general::Additive">Additive</a>, <a class="struct" href="../../alga/general/struct.Multiplicative.html" title="struct alga::general::Multiplicative">Multiplicative</a>> for <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.i16.html">i16</a></code><a href='#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i16' class='anchor'></a><a class='srclink' href='../../src/alga/macros.rs.html#102' title='goto source code'>[src]</a></h3><div class='impl-items'></div><h3 id='impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i32' class='impl'><code class='in-band'>impl <a class="trait" href="../../alga/general/trait.AbstractRing.html" title="trait alga::general::AbstractRing">AbstractRing</a><<a class="struct" href="../../alga/general/struct.Additive.html" title="struct alga::general::Additive">Additive</a>, <a class="struct" href="../../alga/general/struct.Multiplicative.html" title="struct alga::general::Multiplicative">Multiplicative</a>> for <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.i32.html">i32</a></code><a href='#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i32' class='anchor'></a><a class='srclink' href='../../src/alga/macros.rs.html#102' title='goto source code'>[src]</a></h3><div class='impl-items'></div><h3 id='impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i64' class='impl'><code class='in-band'>impl <a class="trait" href="../../alga/general/trait.AbstractRing.html" title="trait alga::general::AbstractRing">AbstractRing</a><<a class="struct" href="../../alga/general/struct.Additive.html" title="struct alga::general::Additive">Additive</a>, <a class="struct" href="../../alga/general/struct.Multiplicative.html" title="struct alga::general::Multiplicative">Multiplicative</a>> for <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.i64.html">i64</a></code><a href='#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-i64' class='anchor'></a><a class='srclink' href='../../src/alga/macros.rs.html#102' title='goto source code'>[src]</a></h3><div class='impl-items'></div><h3 id='impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-isize' class='impl'><code class='in-band'>impl <a class="trait" href="../../alga/general/trait.AbstractRing.html" title="trait alga::general::AbstractRing">AbstractRing</a><<a class="struct" href="../../alga/general/struct.Additive.html" title="struct alga::general::Additive">Additive</a>, <a class="struct" href="../../alga/general/struct.Multiplicative.html" title="struct alga::general::Multiplicative">Multiplicative</a>> for <a class="primitive" href="https://doc.rust-lang.org/nightly/std/primitive.isize.html">isize</a></code><a href='#impl-AbstractRing%3CAdditive%2C%20Multiplicative%3E-for-isize' class='an
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